n 


I 


University  of  California  •  Berkeley 

THE  THEODORE  P.  HILL  COLLECTION 

of 
EARLY  AMERICAN  MATHEMATICS  BOOKS 


PRACTICAL    TREATISE 


ON  THE 


DIFFERENTIAL  AND  INTEGRAL 

CALCULUS, 

WITH   SOME  OF   ITS 

APPLICATIONS    TO    MECHANICS    AND 
ASTRONOMY. 


BY 

WILLIAM   G.   PECK,  PH.D.,  LL.D., 

Profeswr  of   Mathematics   and   Astronomy  in    Columbia.    Colleen 
and  of  Mechanics  in   tht  School  of  Mines. 


A.    S.    BARNES    &    COMPANY, 
NEW  YORK   AND  CHICAGO. 


PUBLISHERS'     NOTICE. 

PECK'S     MATHEMATICAL     SERIES. 

CONCISE,  CONSECUTIVE,  AND  COMPLETE. 


I.  FIRST    LESSONS    IN    NUMBERS. 
II.  MANUAL    OF    PRACTICAL    ARITHMETIC. 

III.  COMPLETE    ARITHMETIC. 

IV.  MANUAL    OF    ALGEBRA. 
V.  MANUAL    OF    GEOMETRY. 

VI.  TREATISE    ON    ANALYTICAL    GEOMETRY. 
VII.  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 
VIII.  ELEMENTARY  MECHANICS  (without  the  Calculus). 
XL  ELEMENTS  OF  MECHANICS  (with  the  Calculus). 


NOTE. — Teachers  and  others,  discovering  errors  in  any  of 
the  above  works,  will  confer  a  favor  by  communicating  them 
to  us. 


Entered  according  to  Act  of  Congress,  in  the  year  1870,  by 

WILLIAM    G.    PECK, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


PREFACE. 


THE  following  pages  are  designed  to  embody  the  course 
of  "  Calculus  and  its  Applications,"  as  now  taught  in 
Columbia  College.  This  course  being  purely  optional,  is 
selected  by  those  only  who  have  a  taste  for  mathematical 
studies,  and  it  is  pursued  by  them  more  with  reference  to 
its  utility  than  as  a  means  of  mental  discipline.  These 
circumstances  have  given  to  the  present  work  a  practical 
character  that  can  hardly  fail  to  commend  it  to  those  who 
study  the  Calculus  for  the  advantages  it  gives  them  in 
the  solution  of  scientific  problems.  To  meet  the  wants 
of  this  class  of  students,  great  care  has  been  taken  to  avoid 
superfluous  matter;  the  definitions  have  been  revised  and 
abbreviated;  the  demonstrations  have  been  simplified  and 
condensed ;  the  rules  and  principles  have  been  illustrated 
and  enforced  by  numerous  examples,  so  chosen  as  to  famil- 
iarize the  student  with  the  use  of  radical  and  transcen- 
dental quantities;  and  finally,  the  manner  of  applying  the 
Calculus  has  been  exemplified  by  the  solution  of  a  variety 
of  problems  in  Mechanics  and  Astronomy. 

The  method  employed  in  developing  the  principles  of 
the  science  is  essentially  that  of  Leibnitz.     This  method, 


4:  PKEFACE. 

generally  known  as  the  method  of  infinitesimals,  has  been 
adopted  for  several  reasons:  first,  it  is  the  method  adopted 
in  all  practical  investigations;  second,  it  is  the  method 
most  easily  explained  and  most  readily  comprehended ; 
and  third,  it  is  a  method,  as  will  be  shown  in  the  final 
note,  identical  in  results  with  the  more  commonly  adopted 
method  of  limits,  differing  from  it  chiefly  in  its  phraseology 
and  in  the  simplicity  of  its  results. 

The  author  cannot  conclude  this  prefatory  note  with- 
out acknowledging  his  obligations  to  those  students  who, 
from  year  to  year,  have  been  willing  to  turn  aside  from 
the  attractive  pursuit  of  classical  learning  to  engage  in  the 
sterner  study  of  those  processes  that  have  contributed  so 
much  to  the  progress  of  modern  science.  Without  their 
interest  and  co-operation  this  book  would  never  have  been 
written.  He  would  also  take  this  opportunity  to  express 
his  thanks  to  his  distinguished  colleague,  Professor  J.  H. 
VAN  AMRISTGE,  not  only  for  many  valuable  suggestions 
made  during  the  progress  of  the  work,  but  also  for  much 
effective  labor  in  reading  and  correcting  the  proofs  as  they 
came  from  the  press. 

COLUMBIA  COLLEGE, 

Nwember  24&,  1870. 


CONTENTS. 


PART   1.-DIFFERENTIAL  CALCULUS. 

I.  DEFINITIONS  AND  INTRODUCTORY  REMARKS. 
Art. 

1.  Classification  of  Quantities 9 

2.  Functions  of  one  or  more  Variables 9 

3.  Geometrical  Representation  of  a  Function 10 

4.  Differentials  and  Differentiation 10 

5.  Geometrical  Illustration 13 

6.  Infinites  and  Infinitesimals 13 

7.  General  Method  of  Differentiation 14 

II.  DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS. 

8.  Definition  of  an  Algebraic  Function 15 

9.  Differential  of  a  Polynomial 15 

10.  Differential  of  a  Product 16 

11.  Differential  of  a  Fraction 17 

12.  Differential  of  a  Power 18 

13.  Differential  of  a  Radical 18 

III.  DIFFERENTIATION  OF  TRANSCENDENTAL  FUNCTIONS. 

14.  Definition  of  a  Transcendental  Function 24 

15.  Differential  of  a  Logarithm 25 

16.  Differential  of  an  Exponential  Function 26 

17.  Differentials  of  Circular  Functions 29 

18.  Differentials  of  Inverse  Circular  Functions 32 

IV.  SUCCESSIVE  DIFFERENTIATION  AND  DEVELOPMENT  OF 
FUNCTIONS. 

1 9.  Successive  Differentials 35 

20.  Successive  Differential  Coefficients 35 

21.  McLaurin's  Formula 37 

M.  Taylor's  Formula 41 


0  CONTEXTS. 

V.  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES  AND  OF 
IMPLICIT  FUNCTIONS. 

Art.  PACK 

23.  Geometrical  Representation  of  a  Function  of  two  Variables.  45 

24.  Differential  of  a  Function  of  two  Variables 4G 

25.  Notation  for  Partial  Differentials 47 

26.  Successive  Differentiation  of  Functions  of  two  Variables  ...  48 

27.  Extension  to  three  or  more  Variables 50 

28.  Definition  of  Explicit  and  Implicit  Functions 50 

29.  Differentials  of  Implicit  Functions 50 

PART  II— APPLICATIONS   OF  THE  DIFFERENTIAL 
CALCULUS. 

I.  TANGENTS  AND  ASYMPTOTES. 

30.  Geometrical  Representation  of  first  Differential  Coefficient. .  53 

31 .  Applications 54 

32.  Equations  of  the  Tangent  and  Normal 55 

33.  Asymptotes 57 

34.  Order  of  Contact 59 

II.  CURVATURE. 

35.  Direction  of  Curvature GO 

36.  Amount  of  Curvature 62 

d7.  Osculatory  Circle 63 

38.  Radius  of  Curvature 64 

39.  Co-ordinates  of  Centre  of  Curvature 65 

40.  Locus  of  the  Centre  of  Curvature 66 

41.  Equation  of  the  Evolute  of  a  Curve 67 

III.  SINGULAR  POINTS  OF  CURVES. 

42.  Definition  of  a  Singular  Point 08 

43.  Points  of  Inflexion 68 

44.  Cusps 70 

45.  Multiple  Points 71 

46.  Conjugate  Points 73 

IV.  MAXIMA  AND  MINIMA. 

47.  Definitions  of  Maximum  and  Minimum 74 

48.  Analytical  Characteristics 75 

49.  Maxima  and  Minima  of  a  Function  of  two  Variables 86 


CONTENTS.  7 

V.  SINGULAR  VALUES  OP  FUNCTIONS. 

Art.  PAOR 

50.  Definition  and  Method  of  Evaluation 88 

VI.  ELEMENTS  OF  GEOMETRICAL  MAGNITUDES. 

61.  Differentials  of  Lines,  Surfaces,  and  Volumes 91 

VII.  APPLICATION  TO  POLAR  CO-ORDINATES. 

52.  General  Notions,  and  Definitions 93 

53.  Useful  Formulas 94 

54.  Spirals 97 

55.  Spiral  of  Archimedes 98 

VIII.  TRANSCENDENTAL  CURVES. 

56.  Definition 99 

57.  TheCycloid 99 

58.  The  Logarithmic  Curve 101 

PART  III.— INTEGRAL  CALCULUS. 

59.  Object  of  the  Integral  Calculus 102 

60.  Nature  of  an  Integral 1 02 

61.  Methods  of  Integration ;  Simplifications 104 

62.  Fundamental  Formulas 104 

63.  Integration  by  Parts 114 

04.  Additional  Formulas ' .  115 

65.  Rational  and  Entire  Differentials 120 

06.  Rational  Fractions 121 

67.  Integration  by  Substitution,  and  Rationalization 128 

68.  When  the  only  Irrational  Parts  are  Monomial 128 

69.  Binomial  Differentials 130 

70.  Integration  by  Successive  Reduction 133 

71.  Certain  Trinomial  Differentials 141 

72.  Integration  by  Series 144 

73.  Integration  of  Transcendental  Differentials. 145 

74.  Logarithmic  Differentials 145 

75.  Exponential  Differentials 147 

70.  Circular  Differentials 148 

77.  Integration  of  Differential  Functions  of  two  Variables 153 

PART  IV.— APPLICATIONS  OF  THE  INTEGRAL   CAL- 
CULUS. 
I.  LENGTHS  OF  PLANE  CURVES. 

78.  Rectification. .  156 


CONTENTS. 
II.  AREAS  OP  PLANE  CURVES. 

Art'  PAOl 

79.  Quadrature 101 

III.  AREAS  OF  SURFACES  OF  REVOLUTION. 

80.  Surfaces  Generated  by  the  Revolution  of  Plane  Curves 105 

IV.  VOLUMES  OF  SOLIDS  OF  REVOLUTION. 

81.  Cubature 1G7 

PART  V.— APPLICATIONS  TO  MECHANICS  AND 
ASTRONOMY. 

I.  CENTRE  OF  GRAVITY. 

82.  Principles  Employed 171 

83.  Centre  of  Gravity  of  a  Circular  Arc 173 

84.  Centre  of  Gravity  of  a  Parabolic  Area 174 

85.  Centre  of  Gravity  of  a  Semi-Ellipsoid  of  Revolution 175 

86.  Centre  of  Gravity  of  a  Cone 176 

87.  Centre  of  Gravity  of  a  Paraboloid  of  Revolution 176 

II.  MOMENT  OF  INERTIA. 

88.  Definitions  and  Preliminary  Principles 176 

89.  Moment  of  Inertia  of  a  Straight  Line 177 

90.  Moment  of  Inertia  of  a  Circle 178 

01.  Moment  of  Inertia  of  a  Cylinder 179 

92.  Moment  of  Inertia  of  a  Sphere 180 

III.  MOTION  OF  A  MATERIAL  POINT. 

93.  General  Formulas 181 

94.  Uniformly  Varied  Motion 182 

95.  Bodies  Falling  under  influence  of  Constant  Force 183 

96.  Bodies  Falling  under  action  of  Variable  Force 186 

97.  Vibration  of  a  Particle  of  an  Elastic  Medium 188 

98.  Curvilinear  Motion  of  a  Point 190 

99.  Velocity  of  a  Point  rolling  down  a  Curve 192 

100.  The  Simple  Pendulum 193 

101.  Attraction  of  Homogeneous  Spheres 195 

102.  Oibital  Motion 199 

103.  Law  of  Force 204 

104.  Note  on  the  Methods  of  the  Calculus 205 


PART  I. 

DIFFERENTIAL  CALCULUS. 

I.  DEFINITIONS  AND  INTRODUCTORY  REMARKS, 

Classification  of  Quantities. 

I.  THE  quantities  considered  in  Calculus  are  of  two 
kinds:  constants,  which  retain  a  fixed  value  throughout 
the  same  discussion,  and  variables,  which  admit  of  all  pos- 
sible values  that  will  satisfy  the  equations  into  which  they 
enter.  The  former  are  usually  denoted  by  leading  letters 
of  the  alphabet,  as  a,  I,  c,  etc.;  and  the  latter  by  final 
letters,  as  x,  y,  z,  etc. ;  particular  values  of  variable  quan- 
tities are  denoted  by  writing  them  with  one  or  more  dashes, 
as  x',  y",  z'",  etc. 

Functions  of  one  or  more  Variables. 

2.  Relations  between  variables  are  expressed  by  equa- 
tions. In  an  equation  between  two  variables,  values  may 
be  assigned  to  one  at  pleasure;  the  resulting  equation  de- 
termines the  corresponding  values  of  the  other.  The  one 
to  which  arbitrary  values  are  assigned  is  called  the  inde- 
pendent variable,  and  the  remaining  one  is  said  to  be  a 
function  of  the  former.  If  an  equation  contain  more  than 
two  variables,  all  but  one  are  independent,  and  that  one  is 
a  function  of  all  the  others.  The  fact  that  a  quantity 


10  DIFFEEENTIAL   CALCULUS. 

depends  on  one  or  more  variables  may  be  expressed  as 
follows : 

y  =f(z) ;  z  -  <p(x,  y) ;  F(x,  y,  z)  =  0. 

The  first  shows  that  y  is  a  function  of  x,  the  second  that 
z  is  a  function  of  x  and  y,  and  the  third  that  x,  y,  and  z, 
depend  on  each  other,  without  pointing  out  which  is  u 
function  of  the  other  two. 


Geometrical  representation  of  a  Function. 

3.  Every  function  of  one  variable  may  be  represented  b} 
the  ordinate  of  a  curve,  of  which  the  variable  is  the  cor- 
responding abscissa.     For,  let  y  be  a  function  of  x,  and 
suppose  x  to  increase  by  insensible  gradations  from  —  oo 
to  +  oo .     For  each  value  of  x  there  will  be  one  or  more 
values  of  y,  and  these,  if  real,  will  determine  the  position 
of  a  point  with  respect  to  two  rectangular  axes.     These 
points  make  up  a  curve,  at  every  point  of  which  the  rela- 
tion between  the  ordinate  and  abscissa  is  the  same  as  that 
between    the   function   and   independent  variable.      This 
curve  is  called  the  curve  of  the  function. 

For  values  of  x  that  give  imaginary  values  of  y  there  are 
no  points;  for  those  that  give  more  than  one  real  value  of 
y  there  is  a  corresponding  number  of  points. 

In  a  similar  manner  -it  may  be  shown  that  a  function 
of  two  variables  represents  the  ordinate  of  a  surface  of 
which  the  variables  are  corresponding  abscissas. 

Differentials  and  Differentiation. 

4.  Of  two  quantities,  that  is  the  less  whose  value   is 
nearer  to  —  oo,  and   that  is   the  greater  whose  value  is 
nearer  to  +  oo.     A  quantity  is  said  to  increase  when  it 


DEFINITIONS   AND   INTRODUCTORY    REMARKS.          11 

approaches  +  GO,  and  to  decrease  when  it  approaches  —  oo. 
The  ordi nates  of  a  curve  originate  from  the  axis  of  X  and 
of  any  two,  that  is  the  greater  whose  extremity  is  nearer 
4-  oo,  and  that  is  the  less  whose  extremity  is  nearer  —  co; 
in  like  manner  of  two  abscissas,  that  is  the  greater  whose 
extremity  is  nearer  -f-  oo ,  and  that  the  less  whose  extremity 
is  nearer  —  oo . 

In  wrhat  follows  we  shall  suppose  the  independent  varia- 
ble to  increase  by  the  continued  addition  of  a  constant  but 
infinitely  small  increment.  For  every  change  in  the  value 
of  the  variable  there  is  a  corresponding  change  in  the  value 
of  the  function.  In  some  cases,  as  the  variable  increases, 
the  function  increases ;  it  is  then  said  to  be  an  increasing 
function :  in  other  cases  the  function  decreases  as  the 
variable  increases ;  it  is  then  said  to  be  a  decreasing  func- 
tion. In  all  cases,  the  change  in  value  is  called  an  incre- 
ment;  for  increasing  functions  the  increment  is  positive, 
and  for  decreasing  functions  it  is  negative.  The  increment 
of  the  function  is  always  infinitely  small,  but  it  is  not  con- 
stant, except  in  particular  cases. 

The  infinitely  small  increment  of  the  independent  varia- 
ble is  called  the  differential  of  the  variable,  and  the  cor- 
responding increment  of  the  function  is  called  the  differ- 
ential of  the  function.  Hence,  the  differential  of  a  quantity 
is  the  difference  between  two  consecutive  values  of  that 
quantify.  It  is  to  be  observed  that  the  difference  is  always 
found  by  taking  the  first  value  from  the  second. 

The  operation  of  finding  a  differential  is  called  differen- 
tiation. The  object  of  the  differential  calculus  is  to  ex- 
plain the  methods  of  differentiating  functions. 


DIFFERENTIAL   CALCULUS. 


Geometrical   Illustratiou. 

5.  Let  KL  be  a  curve  in  the  plane  of  the  rectangular 
axes  OX  and  OY9  and  let  OA  and  OB  be  two  abscissas 
differing  from  each  other  by  an  infi- 
nitely small  quantity  A  B.  Through 
A  and  B  draw  ordinates  to  the 
curve,  and  let  PR  be  parallel  to 
OX.  OA  and  OB  are  consecutive 
abscissas,  AP  and  BQ  are  consecu- 
tive ordinates,  and  P  and  Q  are 
consecutive  points  of  the  curve. 
The  part  of  the  curve  PQ  does 
not  differ  sensibly  from  a  straight  line,  and  if  it  be  pro 
longed  toward  T,  the  line  PT  is  tangent  to  the  curve  at 
P.  If  we  denote  any  abscissa  OA  by  x,  and  the  corre- 
sponding ordinate  by  y,  we  have, 


Fig.  ]. 


The  line  AB  is  the  differential  of  the  independent  varia- 
ble, denoted  by  the  symbol  dx;  RQ  is  the  differential  of 
the  function,  denoted  by  dy\  and  PQ  is  the  differential 
of  the  curve  KL,  denoted  by  ds. 

The  right  angled  triangle,  RPQ,  gives  the  relation 
ds  =  Vdx*  +  dy\  Denoting  the  angle  RPQ  by  6,  we 
have,  from  trigonometry, 


dy 

—  ~ ; 
dx 


_dy_ 
~  ds~ 


dy 


dx 

and,  cosd  =•-=-  = 
cfe 


dx 


-f-  dy* 


DEFINITIONS    AND   INTRODUCTORY    REMARKS.  13 


Infinites  and  Infinitesimals. 

6.  A  quantity  is  infinitely  great  with  respect  to  another, 
when  the  quotient  of  the  former  by  the  latter  is  greater 
than  any  assignable  number,  and  infinitely  small  with  re- 
spect to  it,  when  the  quotient  is  less  than  any  assignable, 
number.  If  the  term  of  comparison  infinite,  quantities  of 
the  former  class  are  called  infinites,  and  those  of  the  latter 
infinitesimals. 

Infinites  and  infinitesimals  are  of  different  orders.  Let 
us  assume  the  series, 

a    a    a 

'  '   ?'  a5'  "?  a'  aX)       '          ' 

In  which  a  is  a  finite  constant  and  x  variable.  If  we 
suppose  x  to  increase,  the  terms  preceding  a  will  diminish, 
and  those  following  it  will  increase ;  when  x  becomes  greater 

than  any  assignable  quantity,  —  becomes  infinitely  small 

30 

•with  respect  to  a,  and  because  each  term  bears  the  same 
relation  to  the  one  that  follows  it,  every  term  in  the  series 
is  infinitely  small  with  respect  to  the  following  one,  and 
infinitely  great  with  respect  to  the  preceding  one.  The 
quantity  ax  being  infinitely  great  with  respect  to  a  finite 
quantity,  is  called  an  infinite  of  i\\.Q  first  order;  ax*,  ax*, 
etc.,  are  infinites  of  the  second,  third,  etc.,  orders.  The 

quantity  -  being  infinitely  small  with  respect  to  a  finite 
x 

quantity  is  called  an  infinitesimal  of  the  first  order;  -^  -3, 

x     x 

etc,  are  infinitesimals  of  the  second,  third,  etc.,  orders. 

It  is  to  be  observed  that  the  product  of  two  infinitesi- 
mals of  the  first  order,  is  an  infinitesimal  of  the  second 


1*  D1FFEKEXT1AJ.   CALCULUS. 

order.  For,  let  x  and  y  be  infinitely  small  with  respect 
to  1,  we  shall  have, 

1  :  x  : :  y  :  xy. 

Hence,  xy  bears  the  same  relation  to  y  that  x  does  to  1, 
that  is,  it  is  infinitely  small  with  respect  to  an  infinitesimal 
of  the  first  order;  it  is  therefore  an  infinitesimal  of  the 
second  order.  The  product  of  three  infinitesimals  of  the 
first  order  is  an  infinitesimal  of  the  third  order,  and  so  on. 
In  general,  the  product  of  an  infinitesimal  of  the  ?/?th  order 
by  one  of  the  nth  order,  is  an  infinitesimal  of  the  (m  +  n)tb 
order.  The  product  of  a  finite  quantity  by  an  infinitesimal 
of  the  wth  order,  is  an  infinitesimal  of  the  wth  order. 

From  the  nature  of  an  infinite  quantity,  its  value  will 
not  be  sensibly  changed  by  the  addition  or  subtraction  of 
a  finite  quantity.  A  finite  quantity  may  therefore  be  dis- 
regarded in  comparison  with  an  infinite  quantity.  For 
a  like  reason  an  infinitesimal  may  be  disregarded  in  com- 
parison with  a  finite  quantity,  or  with  an  infinitesimal  of  a 
lower  order.  Hence,  whenever  an  infinitesimal  is  con- 
nected, by  the  sign  of  addition,  or  subtraction,  with  a  finite 
quantity,  or  with  an  infinitesimal  of  a  lower  order,  it  may 
be  suppressed  without  affecting  the  value  of  the  expression 
into  which  it  enters. 

General  method  of  Differentiation. 

7.  In  order  to  find  the  differential  of  a  function,  we  give 
to  the  independent  variable  its  infinitely  small  increment, 
and  find  the  corresponding  value  of  the  function;  from 
this  we  subtract  the  preceding  value  and  reduce  the  result 
to  its  simplest  form;  we  then  suppress  all  infinitesimals 
which  are  added  to,  or  subtracted  from,  those  of  a  lowei 
order,  and  the  result  is  the  differential  required. 


DIFFERENTIATION   OF   ALGEBRAIC   FUNCTIONS.          15 

This  method  of  proceeding  is  too  long  for  general  use, 
and  is  only  employed  in  deducing  rules  for  differentiation. 

II.  DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS. 

Definition  of  an  Algebraic  Function.     ' 

8.  An  algebraic  function  is  one  in   which  the  relation 
between  the  function  and  its  variable  can  be  expressed  by 
the  ordinary  operations  of  algebra,  that  is,  addition,  sub- 
traction, multiplication,  division,  formation  of  powers  de- 
noted by  constant  exponents,  and  extraction  of  roots  indicated 
by  constant  indices.    Thus, 

y*  =  2px,  y  =  ax3  —  ^/bx,  and  ^/y  =  *^a*x  —  bx*% 
are  algebraic  functions. 

Differential  of  a  Polynomial. 

9.  Let  a  and  c  be  constants,  and  r,  s,  t,  functions  of  x\ 
assume 

y  =  ar  +  s  —  t  +  c\ (1) 

in  which  y  denotes  the  polynomial  in  the  second  member, 
and  is  therefore  a  function  of  x. 

If  we  give  to  x  the  increment  dx,  the  functions  y,  r,  s* 
and  t,  receive  corresponding  increments  dy,  dr,  ds,  dt,  and 
we  have, 

y  +  dy  =  a(r  +  dr)  +  (s  4-  ds)  -  (t  +  dt)  +  c (2) 

subtracting  (1)  from  (2),  we  have, 

dy  =  adr  +  ds  —  dt (3) 

Hence,   to   differentiate   a  polynomial,   differentiate   each 
term  separately  and  take  the  algebraic  sum  of  the  results. 

Comparing  (1)  and  (3)  we  see,  first,  that  a  constant  factor 
remains  unchanged,  and  secondly,  that  a  constant  term 
disappears  by  differentiation. 


16  DIFFERENTIAL   CALCULUS. 

Differential  of  a  Product. 

10.  Let  r  and  s  be  functions  of  x.  Placing  their  pro- 
duct equal  to  y,  we  have, 

y  =  rs (1) 

Giving  to  x  the  increment  dx,  we  have,  as  before, 
y  4-  dy  =  (r  -f  dr)  (s  4-  ds)  =  rs  +  rds  +  sdr  +  drds.  .  (2) 
Subtracting  (1)  from  (2)  and  suppressing  ^TY?S,  which  is  an 
infinitesimal  of  tbe  second  order,  we  have,  after  replacing 
y  by  its  value  rs, 

d(rs)  =  rds  +  sdr (3) 

Hence,  to  differentiate  the  product  of  two  functions, 
multiply  each  by  the  differential  of  the  other,  and  take  the 
algebraic  sum  of  the  results. 

If  we  suppose  r  =  tw,  we  have,  from  the  rule, 

dr  =  tdw  +  wdt ; 
which  substituted  in  (3),  gives, 

d(stiu)  =  twds  +  stdiv  +  swdt (4) 

In  like  manner  the  principle  may  be  extended  to  the 
product  of  any  number  of  functions.  Hence,  to  differen- 
tiate the  product  of  any  number  of  functions,  multiply  the 
differential  of  each  ~by  the  continued  product  of  all  the 
others,  and  take  the  algebraic  sum  of  the  results. 

COR. — If  we  divide  both  members  of  (4)  bystw,  we  have, 
d(sfw)  _  ds  dt  dw  .„. 

•     ••••     1  fJ  I 

stw          s        t         w 

and  similarly,  where  there  are  a  greater  number  of  factors. 
Hence,  the  differential  of  a  product  divided  ly  that  pro- 
duct, is  equal  to  the  sum  of  the  quotients  obtained  ly 
dividing  the  differential  of  each  factor  by  that  factor. 


DIFFERENTIATION   OF   ALGEBRAIC   FUNCTIONS.          17 

Differential  of  a  Fraction. 
11.  Let  61  and  t  be  functions  of  x,  and  assume, 


Giving  to  x  the  increment  dx,  we  have, 


Subtracting  (1)  from  (2), 

+  ds      s      ids  —  sdt 


dy  = 


t  +  dt      t        f  +  tdt ' 

s 


Replacing  y  by  its  value  -,  and  suppressing  tdt  in  compari- 
son with  f ,  we  have, 

ids  -  sdt 
= jr .(8) 

Hence,  the  differential  of  a  fraction  is  equal  to  the  denom- 
inator into  the  differential  of  the  numerator,  minus  the 
numerator  into  the  differential  of  the  denominator,  divided 
by  the  square  of  the  denominator. 

If  either  term  of  the  fraction  be  constant,  its  differential 
will  be,  0.  When  the  denominator  is  constant,  formula  (3) 
becomes, 


'6)  =  7' 


...  (4) 
when  the  numerator  is  constant  it  becomes, 


18  DIFFERENTIAL  CALCULUS. 

Differential  of  a  Power. 
12.  Let  s  be  a  function  of  x,  and  m  a  constant.    Assume 

y  =  *m  .....  (i) 

Giving  to  x  the  increment  clx,  we  have, 

y  +  dy  =  (s  +  ds)m  =  sm  +  msm"lds 
(etc.)  .....  (2) 


In  which  all  the  terms  of  the  development,  after  the  second, 
contain  the  square,  or  some  higher  power,  of  ds.  Subtract- 
ing (1)  from  (2),  we  have, 

dy  =  msm~^ds  +  (etc.)  ds2. 

Suppressing  all  the  terms  of  the  second  member,  after  the 
first,  in  accordance  with  the  principle  laid  down  in  Art.  6, 
and  replacing  y  by  its  value,  we  have, 

d(sm)  =  msm~^ds  .....  (3) 

Hence,  to  differentiate  any  power  of  a  function,  diminish 
the  exponent  oj  'the  function  by  1,  and  multiply  the  result  by 
the  primitive  exponent  and  the  differential  of  the  function. 


Differential  cf  a  Radical. 
13.  Let  s  be  a  function  of  x,  and  assume, 

y  =  ft  .....  (i) 


• 

DIFFERENTIATION"   OF   ALGEBRAIC   FUNCTIONS.          19 


We  may  write  (1)  in  the  form, 
1 


Differentiating  (2),  as  in  the  last  article,  we  have, 

1-1  1-rc 

dy  =  -sn     ds  =  -s  n  ds  .....  (3) 

J      n  n 

Replacing  y  by  its  value,  and  remembering  that 
s\—n  —     —    We  have. 


. 

(4) 


That  is,  the  differential  of  a  radical  of  the  ^th  degree 
5.3  equal  to  the  differential  of  the  quantity  under  the 
radical  sign,  divided  by  n  times  the  (n  —  l)th  poiver  of  the 
radical  (ifcz  *ra, 


uuj  -nC 
, 


^~s  =  f-  .....  (5) 

oT?  "  ^     1& 

That  is,  the  differential  of  the  square  root  of  a  quantity 
is  equal  to  the  differential  of  the  quantity  under  the  radical 
sign,  divided  by  twice  the  radical.  &  b*-^ 

The  preceding  rules  are  sufficient  to  differentiate  any 
algebraic  function  whatever. 


20  DIFFERENTIAL   CALCULUS. 

EXAMPLES. 

1     1.  Let  y  =  5z3. 

This  is  the  product  of  the  function  cc3  by  the  constant  5 ; 
hence,  (Art.  9), 

dy  =  5d(x*)  =  5X  3x*dx  =  15x*dx (Art.  12). 

2.  Let  y  =  x3  +  2x2  +  3x  +  4. 

This  is  a  polynomial.     Hence,  from  Art.  9, 
dy  =  3x2dx  +  ^xdx  +  MX. 

3.  Lety  =  («  +  J»)s. 

Considering    a  +  Jo;     as   a   single  quantity,  we  have, 
(Art.  12), 

dy  =  3(«+ te)8  d(a  +  lx)  =  3(a  +  lx)*bdx  =  3b(a  +  bx)*dx. 

4.  Let  y  =  (a  +  lx*)n. 
dy  =  n(a 


5.  Let  y  =  A/«2  -£2  =  (a2  -  z2)2. 

dy  =  J(a2  -  z2)"^  x  -  Zxdx  =  - 

6.  Let  y  =  (a  +  x)(b  +  2a?»). 
By  Art.  10,  we  have, 

dy=  (a  +  x)d(b  +  2x2)  +  (b  +  2x2)d(a  -\- 
i-  (b  +  2x*)dx ;  or,  «fy 
B  +  b)dx. 


DIFFERENTIATION   OF   ALGEBRAIC   FUNCTIONS.          21 


• 

By  Art.  11,  we  have, 


x2)d(ax)  —  axd(a*  +  a2)  __  a(a2-x2) 

~~ 


, 


8.  Let  y=^x+ 
dy  =  d[x  +  (a2 

X 
W,t,  d\x  +  (a* 


10.  y=a»       —  to       ±<i 

Am.  dy  =  - 

11.  y  =  (a2  -  x*)*.  Ans.  dy  =  -  ±x(a*  -  z*)dx. 

12.  y  =  (%ax  —  x2)3.  Ans.  dy  =  6(a  -  x)(2ax  —  x*)'dx, 


DIFFERENTIAL   CALCLLUS. 


13.  y  =  (a+bxn).  Ans.dy  =  mn1)xn-l( 

14.  y=(2ax  +  x*)n.  Ans.dy=2n(a  +  x)(2ax 


15.  y  =  x(a  +  x)(az 

Ans.  dy  =  (a3  -f  2a*x  +  Sax2  +  ±x*)dx. 

16.  y  =  (a  +  x)m(b  +  x}n. 


17.  v  = 


a  —  x 
a  +  x' 


18.  y  =¥• 


x  +  3 
-x  +  l 


19.  y  =fi 

20.  y  = 

21.  y  = 

22.  y  = 

23.  y  =  (a  —  a;) 

24.  y=  (a  +  x)Va  — 


+OX 


Ans.  dv  =  — 


(x  +  2)(x 

( 


.  dy  =  7- 

»  - 


-  2 


Ans.  dy  = 


Ans.  dy  =    xdx- 

' 


(2rcg  +  b)dx 

~  2V ax2  +  bx  +  o; 


(«  +  3:?;)^ 

-4ft*.  tfy  =  —  - — ; — —  . 
%Va  +  x 

(a  -  3x)dx 

Ans.  dy  = ; — - — 

2Va  -  x 


DIFFERENTIATION    OF   ALGEBRAIC    FUNCTIONS.          23 

» 

*  -  3)dx 


.  , 

25.  y  —  \l  —  x*  -{-  Xs)*.    Ans.  dy  =  - 


26.  y  —  (a2  —  x2)Va  +  x. 

Ans.  dy 


87.  y  =  -/=• 
va  —  » 


23.      = 


.  dy  = 


(3a  - 


adx 


20.      =x- 


Ans.  dy  = 


2v/tf*-^i(»-rVat*- 


30.  y  =  x(l  4  x 

31.  y=  (1 

32.  v  =  — 


_*•).   Ans.  dy  = 


_ 

v  1  — 


a  +  VI  — 


33.  y  =  (1  +  x)Vl  -  x.  Ans.  dy  = 


-  a; 


34.      = 


DIFFERENTIAL  CALCULUS. 


36-  y  =      2x  -1-  V%x  -  1  -  V%z  -I-  etc.,  ad  inf. 


Squaring,^8  =2a;-l-V2aj-  1  -  V&e  -  1  -  etc.,  ad  inf. 
Hence,  y*  =  2x  —  1  —  y,  or  y9  +  y  =  Zx  —  1. 


Solving,  y  =  -  5 


/.<&  =  ±5 


III.  DIFFERENTIATION  OF  TRANSCENDENTAL  FUNCTIONS. 

Definition  of  a  Transcendental  Function. 

14.  A  transcendental  function  is  one  in  which  the  rela- 
tion between  the  function  and  its  variable  cannot  be  ex- 
pressed by  the  ordinary  operations  of  algebra. 

Transcendental  functions  are  divided  into  three  classes* 
Logarithmic,  in  which  the  relation  is  expressed  by  loga- 
rithmic symbols;  Exponential,  in  which  the  variable  enters 
an  exponent;  and  Circular,  in  which  the  relation  is  ex- 
pressed by  means  of  trigonometric  symbols.  Thus, 

y  =  log  (ax*  +  #),  is  a  logarithmic  function, 
y  =  (ax2  +  cYx~c,  is  an  exponential  function, 
and  y  =  sin2#  —  2tan#,  is  a  trigonometric  function. 


TRANSCENDENTAL  FUNCTIONS.  25 

Differential  of  a  Logarithm. 
15.  It  is  shown  in  Algebra  that 

/  ?/2          -yS         ?/4  \ 

log  (1  +  y)  =  M\JJ  -J—  +  ^  — '--  +  elc.J  ....  (1) 


In  which  M  is  the  modulus  of  the  system  and  y  any  quan- 
tity whatever. 

Substitute  for  y  the  quantity  —  ,  s  being  a  function  of  #, 

s 

and  we  have,  after  reduction, 

4-  ds         ,ds      ds*       ds* 


But  the  logarithm  of  a  quotient,  is  equal  to  the  logarithm 
of  the  dividend,  diminished  by  the  logarithm  of  the  divi- 
sor; changing  the  form  of  the  first  member  and  suppress- 
ing all  the  terms  in  the  second  member,  after  the  first, 
(Art.  6),  we  have, 

tLt 

log  (s  +  ds)  -  log  s  =  M-  .....  (3) 

s 

But  the  first  member  is  the  difference  between  two  con- 
secutive values  of  log  s;  it  is  therefore  the  differential  of 
the  logarithm  of  s;  hence, 

fj~ 

d(logs)  =  M-  .....  (4) 

s 

That  is,  the  differential  of  the  logarithm  of  a  quantity  is 
equal  to  the  modulus  into  the  differential  of  the  quantity, 
divided  ly  the  quantity. 
In  analysis,  Napierian  logarithms  are  almost  always  used. 


26  DIFFERENTIAL  CALCULUS. 

Denoting  the  Napierian  logarithm  by  I,  and  remembering 
that  the  modulus  of  this  system  is  1,  we  have, 


The  base  of  the  Napierian  system  is  represented  by  the 
letter  e,  and  its  logarithm  in  that  system  is  equal  to  1. 

Differential  of  an  Exponential  Function. 

16.  Let  a  be  a  constant  quantity,  s  any  function  of  z> 
and  assume 

y  =  <f (i) 

Taking  the  logarithms  of  both  members  of  (1),  we  have, 
ly  —  sla (2) 

Differentiating  (2),  we  have, 

dy 

-^-  —  la  X  ds (3) 

y 

"Replacing  y  by  its  value,  and  reducing,  we  have, 
d(as)  =  triads (4) 

Hence,  to  differentiate  a  quantity  formed  by  raising  a  con- 
stant quantity  to  a  power  denoted  by  a  variable  exponent, 
multiply  the  quantity  ly  the  logarithm  of  the  root  and  the 
differential  of  the  exponent. 
Again,  let  us  have, 

y=ts (1) 

in  which  both  t  and  s  are  functions  of  x.    Taking  the 
logarithm  of  both  members,  we  have, 

ly=slt (2) 


•       TRANSCENDENTAL   FUNCTIONS.  27 

\\rheiice,  by  differentiating  both  members, 

f=W?  .....  (3) 

Replacing  y  by  its  value,  clearing  of  fractions  and  reducing, 
we  have, 


(4) 


That  is,  to  differentiate  a  quantity  formed  by  raising  a 
variable  quantity  to  a  power  whose  exponent  is  variable, 
differentiate  first  as  though  the  root  alone  were  variable, 
then  as  though  the  exponent  alone  were  variable,  and  take 
the  sur.i  of  the  results. 

EXAMPLES. 


1.  Let  y  =  l  =  l(a  +  x)-l(a-z); 

(t  —  x 


d(a  +  cc)      d(a  —  x)          dx  dx 

:  ail  =  -  --  —  --  1  -- 

a  +  x  a  —  x         a  +  x      a  —  x 


9,  Let  y  =  I  A/1—  =  ^(1  +  x)  -  \l(l  -  x); 


1    dx         1    dx  dx 

.-.  dy  =  -    —  -  +  a        —  =  r    ~~ 
9      21  +  x      21  —  x      l  —  x 


3.  Txjt  y  ~  (a)e  ;   /.  ly  =  exla,  and  —  =  lae^ledx. 


But,  le  —  1,  /.  dy  =  (a}e  e*ladx.     Ans. 


28  DIFFERENTIAL   CALCULUS. 

4.  Let  y  =  l(lx)  =  T'x. 

* 

d(lx]       dx 

By  the  rule,  dy  =  — \—  =  — r-» 
Ix         xlx 

5.  y  =  #Z(a  4-  #).          ^4^5.  tZ^  —  /(«  4-  #)(/£  4- 


xdx 
a  +  x 


6.  y  =  l(a  +  xY  =  %l(a  4-  x).  Ans.  dy  — 

l.y  =  <?(x-l). 

8.  y  =  e%T2  -  2z  +  2). 


a  4-  x 

Ans.  dy  =  e^atffa;. 
.  dy  -  exxzdx. 


, 
Ans.  dy  —  ----  dx. 


Ans.  dy  =  ex\lx  +  a  k 


11.  ?/  =  /»  +  a  4- 


12.      = 


13.  y  =  I 


+  1  +  a; 


15.  y=aT' 

16.  y=alr. 


Ans.  dy  = 


dx  dx 

Ans.  dy  —  ---  ._ 


,2- 


1  _  ^2 

-4w*.  Jy  — 
Ans.  dy  —  a 


I  +  x  . 
dx. 


TRANSCENDENTAL  FUNCTIONS.  29 

Differentials  of  Circular  Functions. 
17.  Assume  the  equation, 

y  =  siux (1) 

Giving   to  x    the   increment  dx,  and   developing,   we 
have, 

y  +  dy  =  sin  (a;  +  dx) 

=  smxcosdx  +  cosxsmdx  .  .  .  .  (2) 

Because  the  arc  dx  is  infinitely  small,  its  cosine  is  equal 
to  1,  and  its  sine  is  equal  to  the  arc  itself.    Hence. 

y  -f  dy  =  smx  +  cosxdx (3) 

Subtracting  (1)  from  (3),  and  replacing  y  by  its  value, 
we  have, 

d(smx)  =  cosxdx (a) 

Equation  (a)  is  true  for  all  values  of  x\  we  may  there- 
fore replace  x  by  (90°  —  x) ;  this  gives, 

c/[sin(90°  -  a;)]  =  cos(90°  -  a;)c?(90°  -  x) (4) 

Reducing,  we  have, 

d(cosx)  =  —  siuxdz (£) 

We  have,  from  trigonometry,  the  relation, 

sinrc 

tana;  =  -   - (5) 

cosz 

Differentiating  both  members,  we  have, 

,,,       ,       cosxd  (sinx)  —  sinxd  (cosx) 
d(t&nx)  =  -  — s .', 

v  3 


SO  DIFFERENTIAL   CALCULUS. 

Performing  indicated  operations,  and  reducing  by  the 
relation,  sin2  a;  -f  cos8z  —  1,  we  have, 

dx 

c?(tanz)  =  -  —  -  .....  (c) 

1  2 


Replacing  x,  in  formula  (c),  by  (90°  —  x)  and  reducing, 

we  have, 

rlv 

d(cotx)  =  -  -^-  .....  (d) 

Sill2  X  t 

We  have,  from  trigonometry, 

ver-sinz     =  1  —  cosx  .....  (G) 
co-versinz  =  1  —  sinz  .....  (7) 

Differentiating  (6)  and  (7),  we  have, 

c?(ver-siii£c)  =  siuxdx  .....   (e) 
d(co-\GYS\nx)  =  —  cosxdx  .....  (/) 

Wo  have,  from  trigonometry, 

secz  =  -  .....  (8) 
cosz 

cosecx  =  -:  —  .....  (0) 
suix 

Diflerentiating  (8)  and  (9),  we  have,  after  reduction, 


d(secx)  =  tsiiixsecxdx  . 
d(cosQCx)  ==  —  cota;  cosecxdx 


The  lettered  formulas,  from  (a)  to  (/*)  inclusive,  are 
sufficient  for  the  differentiation  of  any  direct  circular 
function. 


TRANSCENDENTAL   FUNCTIONS.  31 

EXAMPLES. 

1.  Let  y  =  sin  mo:. 

dy  =  cosmx  d(?)ix)  =  m  cosmx  .  dxt 

2.  Let  y  =  smmx. 

dy  =  m  sinm~  x  dsmx  =  m  &mm     xcosxdx. 

3.  Let  y  =  sm2x  cosx. 

dy  =  (d  sm2x)  (cosx)  +  (dcosx)(sm2x) 

=  2cos2x  cosx  dx  —  sin2x  smx  dx. 

4.  Let  ?/  =  l(smzx)  —  2lsinx. 

.  dsinx      2cosx  7 

dy  =  3  —:  —  =  ~.  —  dx  =  2cotx  dx, 
smx         smx 


5.  Let  y  =  j  =  ^(i  +  cosx)  -  %l(l  -  cosx). 

V  1  —  cosay 

_        1  smxdx       1    sinxdx  sinxdx  dx 

2  I  +  cosx      21  —  cosx  ~       1—  cos*x  ~       siuz* 

6.  Let  y  =  ex  cosx. 

dy  =  exd(cosx)  +  cosx  d(ex)  —  ex(cosx  —  smx)  dx. 

7.  y  =  z-sinx  +  cos  x.  Ans.  dy  =  xcosxdx. 

8.  y  =  2xsinx  +  (2  —  xz)cosx.          Ans.  dy  =  x2  sinxdx. 

9.  y  =  tana  —  x.  Ans.  dy  =  ttm*xdx. 
10.  y  =  ecosxsmx.            Ans.  dy  =  ecosx(cosx  -  sin*x)dx. 


f  cot^\ 

11.  y  =  2l(&in.x)  +  coseca:.     Ans.  dy  =  [  2cotx  --  :   -jdx, 


32  DIFFERENTIAL   CALCULUS. 

12.  y  =  l(cosx  +  V—  1  sinz).  Ans.  dy  =  V—  idx 


.  dx 

-  -)  Ans.dy  =  --  . 

1  —  smxj  J       cosx 


14.  y  =-  /[tan (45°  +  Jaj)]. 

17 


15.  y  —  sin(fe).  Ans.  dy  =  -cos(lx)dx. 


Differentials  of  Inverse  Circular  Functions. 

18.  It  is  often  convenient  to  regard  an  arc  as  a  function 
of  one  of  its  trigonometrical  lines.  Such  functions  are 
called  inverse  circular  functions,  and  are  expressed  by 
such  symbols  as  the  following : 

sin~~V>  cos   V>  tan   V*^, 

which  are  read  the  arc  ivliose  sine  is  y>  the  arc  whose  cosine 
is  y,  the  arc  whose  tangent  is  y,  etc. 

Formulas  for  the  differentiation  of  inverse  circular  func- 
tions may  be  deduced  from  the  lettered  formulas  of  the  last 
article.  Thus,  from  formula  («),  we  find, 


cosx 
Tf  we  make  sins  =  y,  we  have, 

x  =  sin"1?/,  and  cosx  =  Vl  — 
Substituting  these  in  (1),  we  have, 


TRANSCENDENTAL  FUNCTIONS.  33 

In  like  manner,  from  formula  (b),  we  find, 


smx 
Making  cosa;  =  y,  we  have, 


x 


=  cos~  y,  and  sinz  =  Vl  —  y*  ; 


Substituting  these  in  (2),  we  have, 


vr^ 

By  a  similar  course  of  reasoning  we  find  from  the  re- 
maining lettered  formulas  of  Art.  17,  the  corresponding 
formulas  lor  inverse  circular  functions,  as  given  below: 

_1  dy 


dy 

1  +  y* 


— 1  dy 

d  (versin     y)  = 


-I,A  *y 


d(co  versin    1y)  = —  (/"*) 

dv 
,// — 1..\     •  _        y  /~i\ 


yVy*  -  i 


£n     ••""> 


2* 


34 


DIFFERENTIAL   CALCULUS. 
EXAMPLES. 

Let  y  —  cos~~1#/v/l  —  #8  =  cos" 


-  x  *  = 


and  A/I  _ 


/.  dy  —  — 


(1  - 


Let     =  sin 


1 

~ 


( 
vrrP 


dx 


3.  Let    y  =  sin"1 2x\/l  —  %*• 


2(1  - 


-* 


_ 


= tan 


2dx 


•    DEVELOPMENT   OF  FUNCTIONS.                            35 

«l    /y»  //7    -^—    /F\  & 

6.  y  =  V#2  —  a;2  4-  flrsin"  — .     Ans.  dy  =  I I  dx. 

a  \a  +  x/ 


.   _ 
7.      =  —  versm    A— . 


Vd^ 

8.  y  =  sec-12a;. 


IV.  SUCCESSIVE  DIFFERENTIATION  AND  DEVELOPMENT 
OF  FUNCTIONS. 

Successive  Differentials. 

19.  The  differential  obtained  immediately  from  the  func- 
tion is  the  first  differential  of  the  function  ;  the  differential 
of  the  first  differential  is  the  second  differential  of  the  func- 
tion ;  the  differential  of  the  second  differential  is  the  third 
differential  of  the  function,  and  so  on ;  differentials  thus 
obtained  are  called  successive  differentials,  and  the  opera- 
tion of  obtaining  them  is  called  successive  differentiation. 
If  a  function  be  denoted  by  y,  its  successive  differentials 
will  be  denoted  by  the  symbols  dy,  d*y,  d*y,  etc.     Thus, 
if  y  =  ax3,  we  have,  by  successive  differentiation,  dx  be- 
ing constant,  dy  =  3ax2dx,  dzy  =  Gaxdx2,  d*y  =  6adx3, 
fry  =  0. 

Successive  Differential  Coefficients 

20.  If  the  differential  of  a  function  be  divided  by  the 
differential  of  the  variable,  the  quotient  is  the  first  differ- 
ential coefficient  of  the  function  ;  the  differential  coefficient 
of  the  first  differential  coefficient  is  the  second  differential 
coefficient  of  the  function,  and  so  on.    Differential  coeffi- 


36  DIFFERENTIAL   CALCULUS. 

cients,  derived  in  this  manner,  tire  called  successive  differ- 
ential coefficients.  If  a  function  of  x  be  denoted  by  y, 
its  successive  differential  coefficients  are  denoted  by  the 


symbols,   -r,      ~,     -,  etc.    Thus,  if  we  have,  as  before, 


y  —  ax3,  we  have,  from  the  principles  just  explained, 


It  is  to  be  observed  that  the  successive  differential 
coefficients  are  entirely  independent  of  the  differential  of 
the  variable ;  so  long  therefore  as  this  is  infinitesimal,  the 
differential  coefficients  will  in  no  way  be  affected  by  any 
change  in  its  absolute  value. 

EXAMPLES. 

J^ind  the  successive  differential  coefficients  of  the  follow- 
ing functions: 

1 .  v  =  axn. 


.(     =  nax^;          =  n(n  - 


n(n  -  l)(n  -  2)axn~    ;  etc 

If  n  is  a  positive  whole  number,  there  will  be  a  finite 
number  of  successive  differential  coefficients;  otherwise 
their  number  is  infinite. 

2.  y  =  ax3  f  bx*. 

Ans.  ^  = 
dx 


DEVELOPMENT   OF  FUNCTIONS.  3? 

3.         =  fl*. 


4.  y  =  since. 


Ans.  -/•  —  cosx:  -=-f-  =  —  sin  £  ; 
dx  dx* 


—2.  —  _  cosx:  -5-—  —  smz:  etc. 
c?a;3  ' 

5.  y  =  Z(a?  +  1). 


6.  y  =  xex. 


McLaurin's  Formula. 

21.  McLaurin's  formula,  is  a  formula  for  developing  a 
function  of  one  variable  into  a  series  arranged  according 
to  the  ascending  powers  of  that  variable,  the  coefficients 
being  constant.  Let  y  be  any  function  of  x,  and  assume 
the  development, 


=  A+Bx+  Cx*  +  Dx9  +  Ex*  +  etc  ......  (1) 


38  DIFFERENTIAL   CALCULUS. 

It  is  required  to  find  such  values  for  A,  B,  C,  etc.,  as  will 
make  the  assumed  development  true  for  all  values  of  x. 
Differentiating,  and  finding  the  successive  differential  co- 
efficients of  y,  we  have, 


2Cx  +  Wtf  +  4^3  +  etc.  ...  (2) 
+  l.Z.Wx  +  3.4^2  +  etc.    .  .  .  (3) 


ax 


4-  etc  .......  (4) 

etc.,  etc.,  etc. 

But,  by  hypothesis,  the  value  of  y,  and  consequently  the 
values  of  its  successive  differential  coefficients,  are  to  be 
true  for  all  values  of  x\  hence,  they  must  be  so  for  x  —  0. 
Making  x  ~  0,  in  equations  (1),  (2),  (3),  etc.,  and  denoting 
what  y  becomes  under  that  hypothesis  by  (y}  ;  what 

~  becomes  by  (  —  )  ;  what  •=-?  becomes  by  (  -7-^  j,  and  so 
dx  '  \clxJ  dxz  J  \dx2J 

on  ;  we  have, 

(y)       =A;  .:A  =  (y); 


123-D-     '   Z?= 

T23\3& 

etc.,  etc.,  etc. 


DEVELOPMENT   OF    FUNCTIONS.  39 

Substituting  these  values  in  (I),  we  have, 
dy\     x 


i        »       1       T      n     I  -*     ,TK       i        1       T      o     I    •   -•     ^    r\       I       ^  tt-/«    • 

which  is  the  formula  required.  Hence,  to  develop  a 
function  of  one  variable  in  terms  of  that  variable,  find  its 
successive  differential  coefficients ;  then  make  the  variable 
equal  to  0  in  the  function  and  its  successive  differential 

coefficients,  and  substitute  the  results  for  (y),  (~f\  \~d£i 
etc.,  in  formula  (5). 

Thus,  let  it  be  required  to  develop  sinz  into  a  series 
arranged  with  reference  to  x.    We  have, 

y     =  sinz,         .-.   (y)  =  0 

dy  fdy 

—   =  cosx,        .:   [  -?- 
dx  \dx 


=  —  smz 


dx~*=      *y-  i^J  = 

etc.,  etc.,  etc. 

Hence, 


In  like  manner  we  find, 

°°SX  =l-        +  -  +  etc 


It  is  to  be  observed  that  (7)  may  be  found  from  (6)  by 
differentiating  and  then  dividing  by  dx. 


40  DIFFEKENTIAL  CALCULUS. 

McLau rin's  Formula  enables  us  to  develop  any  function 
of  one  variable  when  it  can  be  developed  in  accordance 
with  the  assumed  law.  But  there  are  functions  which 
cannot  be  developed  according  to  the  ascending  powers  of 
the  variable ;  in  this  case  the  function,  or  some  of  its  suc- 
cessive differential  coefficients,  become  oo,when  the  variable 
is  made  equal  to  0. 

As  a  general  rule,  when  the  application  of  a  formula 
gives  an  infinite  result,  the  formula  is  inapplicable  in  that 
particular  case. 

EXAMPLES. 
1.  y  =  (a  +  X)U. 


Ans 


n   ,    n   n-1     ,  n  (n  —  1)   n-2  . 
.  y  =  a    +y#      x+       12 


n  («-!)(«- 2)  g.-8gi 

JL,</,{) 


X2    ,  X3         X4*        X6 

^  +  ^--  +  - 


3.  y  =  a*. 


4.  y  = 

Make  x*  —  2;,  and  devoir  p;  then  replace  z  by  its  value. 

x2      x*      x6       5x* 


DEVELOPMENT   OF   FUNCTIONS.  41 

5.  y  =  e*™x. 

,  x*          3x*  Sx* 

'=    [+*  +       - 


1.2.3.4.5.6 
6.       ^ 


r-n  +  etc. 


Taylor's    Formula. 

22.  Taylor's  Formula  is  a  formula  for  developing  a 
function  of  the  sum  of  two  variables  into  a  series  arranged 
according  to  the  ascending  powers  of  one,  with  coefficients 
that  are  functions  of  the  other. 

LEMMA. 

If  u'  —  f(x  -\-  y),  the  differential  coefficient  of  u'  will  be 
the  same,  whether  we  suppose  x  to  remain  constant  and  y 
to  vary,  or  y  to  remain  constant  and  x  to  vary,  for  the 
form  of  the  function  is  the  same,  whichever  we  suppose  to 
vary  ;  and  it  has  been  shown,  in  Art.  20,  that  the  value  of 
the  differential  coefficient  is  independent  of  the  value  of 
the  differential  of  the  variable.  Hence,  if  x  +  y  be  in- 
creased either  by  dx  or  by  dy,  and  the  differential  coeffi- 
cient taken,  the  result  will  be  the  same,  which  was  to  be 
shown. 

Let  u'  be  a  function  of  (x  +  y),  and  assume  the  devel- 
opment, 

u'  =  P+Qx  +  Rx*  +  Sx*  +  Tx*  +  etc  ......  (1) 

in  which  P,  Q,  R,  etc.,  are  functions  of  y.     It  is  required 
to  find  such  values  of  P,  Q,  R,  etc.,  as  will  make  equation 


42  DIFFERENTIAL  CALCULUS. 

(1)  true  for  all  values  of  both  x  and  y.  Since  the  assumed 
development  is  to  be  true  for  all  values  of  x  and  t/,  it  must 
be  so  for  x  =  0.  Making  x  =  0  in  (1),  and  denoting  what 
u'  becomes  under  this  hypothesis,  by  u,  we  find, 


That  is,  P  is  what  the  original  function  becomes,  when 
the  leading  variable  is  made  equal  to  0. 

Finding  the  differential  coefficient  of  u',  under  the  sup- 
position that  x  is  constant  and  y  variable,  we  have, 

du'      dP      dQ     t  dR         dS   . 

-=—  =  -=-  +  -T^X  +^-x*  +  -=—  x*  +  etc  ......  (2) 

dy       dy       dy         dy          ay 

Again,  finding  the  differential  coefficient  of  u\  on  the 
supposition  that  y  is  constant  and  x  variable,  we  have, 


-  =  Q  +  2l?x  +  3Sx*  +  ±Tx3  +  etc  ......  (3) 

ux 

But,  the  first  members  of  (2)  and  (3)  are  equal,  by  the 
lemma;  hence  their  second  members  are  also  equal.  If 
we  place  them  equal  we  have  an  identical  equation,  because 
it  is  true  for  all  values  of  x,  and  consequently  the  coeffi- 
cients of  the  like  powers  of  x  in  the  two  members  are 
equal  to  each  other.  Placing  their  coefficients  equal,  we 
have  the  following  results  : 

_dP  .  dn 

V    ~  dy  '  »-  dy9 


~~ 


oo  _  o_ 

=  ~  " 


etc.,  etc.,  etc. 


DEVELOPMENT   OF   FUNCTIONS.  43 

Substituting  the  values  of  P,  Q,  R,  S,  etc.,  in  (1),  we 
have, 


;  _          du    x 

f  dy  '  I  +  dy*  '        +      *  ' 


which  is  the  required  formula.  Hence,  to  develop  any 
function  of  the  sum  of  two  variables,  we  make  the  leading 
variable  equal  to  0,  and  find  the  successive  differential 
coefficients  of  the  result;  then  substitute  them  in  the 
formula. 

Thus,  let  it  be  required  to  develop  (x  +  y}n  into  a 
series  arranged  according  to  the  ascending  powers  of  y. 
Making  y  =  0,  we  have, 

u      =  xn, 


etc.,  etc.,  etc. 

Substituting  these  in  (4),  we  have, 

(x  +  y)n  =  zn  +  ~xn-\j  +  '-iM 


n(n  —  1  (n—  2)  n* 

-     --     --  Lxn    d3  +  etc. 


44  DIFFERENTIAL   CALCULUS. 

This  is  the  binomial  formula,  in  which  n  is  any 
constant. 

The  formula  is  applicable  to  every  function  of  the  sum 
of  two  variables;  but  it  sometimes  happens  that  certain 
values  of  the  variable  entering  the  coefficients  make  the 
first  term  or  some  of  its  successive  differential  coefficients 
infinite  ;  for  these  particular  values,  the  function  cannot 
be  expressed  by  a  series  of  the  proposed  form. 

EXAMPLES. 

Develop  the  following  functions  in  terms  of  y. 

1.  u'  =  sm(x  +  y). 

Making  y  =  0,  we  have,  u  =  sinz  ; 

du  dzu 

./.-=-  =  cosx-j  -j—9  —  —  smx:  etc.; 
dx  dx2 

it  y*  y3 

nence,  u  =  sin#  -f  cosx^-  —  since  ~-  —  cosx-^-r- 

1  l./O  l.-C.d 


Making  x  =  0,  whence  since  =  0,  and  cosx  =  +  1,  we  have, 
n  -  siny  =  y  -  ^  +  1  ^  4  ,  +  etc.,  as  already  shown. 

2.  u  =Z(a?§+y). 

Ans.  u'  =  Ix  +  ¥  -  |^  4-  ~  -  -—  +  etc, 
x      2x*       3x3       4o;4 


3.  w'  = 

Ans.  u'  =  a*(l  +  (la)y  +(^/2  +§^V  +  etc.). 


I M  T  L1C IT   F  U  X  CT 1 0  XS. 


4.  u'  =  cos(x  -f  ?/)• 

/  •    y  v2 

Ans.  u  =  COBX  —  smar   —  cosx  — 


+  SUM  r-n   +  COSX 


V.  DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  VARIABLES, 
AND  OF  IMPLICIT  FUNCTIONS. 

Geometrical  Representation  of  a  Function  of  two  Variables. 

23.  It  may  be  shown,  as  in  Art.  3,  that  every  function 
of  two  variables  represents  the  ordinate  of  a  surface  of 
which  those  variables  are  the  corresponding  abscissas.  If 
the  vertical  ordinate  be  taken  as  the  function,  it  may  be 
expressed  by  the  equation, 


In  this  equation,  x  and  y  are  independent  variables,  and 
each  may  vary  precisely  as  though  the  other  were  constant. 
To  illustrate,  let  PQR 
be   the    surface   whose 
equation  is  (1). 

If,  for  a  given  value 
of  y  as  OS,  we  suppose 
x  to  vary,  equation  (1) 
will  represent  a  section 
of  the  surface  parallel 
to  the  plane  xz  and  at 
the  distance  OB  from 
it.  If  for  a  given  value 
of  x,  as  OA9  we  sup-  Fig>2. 

pose  y  to  vary,  equa- 
tion (1)  will  represent  a  section  parallel  to  the  plane  yi 


46  DIFFERENTIAL   CALCULUS. 

and  at  the  distance  OA  from  it.  These  sections  will  be 
called  parallel  sections,  and  the  corresponding  planes  will 
be  called  planes  of  parallel  section.  The  planes  A  CD 
and  BEF  determine  the  ordinate  KK'  =  z.  Let  A  CD 
A'C'D',  be  consecutive  planes  of  parallel  section,  at  a  dis- 
tance from  each  other  equal  to  dx,  and  BEF,  B'E'F',  also 
consecutive  planes  of  parallel  section,  at  a  distance  from 
each  other  equal  to  dy.  These  planes  determine  four 
ordinates  KK',  LL',  NN',  and  MM',  which  may  be  de- 
noted by  the  symbols  z,  z',  z",  and  z'".  Of  these,  z  and  z' 
are  consecutive  when  x  alone  varies,  z  and  z"  are  consecu- 
tive when  y  alone  varies,  and  z  and  z'"  are  consecutive  when 
both  x  and  y  vary. 

Differentials  of  a  Function  of  two  Variables. 

24.  The  difference  between  two  consecutive  states  of  the 
function  when  x  alone  varies  is  called  the  partial  differ- 
ential with  respect  to  x,  and  is  denoted  by  the  symbol 
(dz)x  ;  the  difference  between  two  consecutive  states  when 

y  alone  varies  is  called  the  partial  differential  with  respect 
to  y,  and  is  denoted  by  the  symbol  (dz)  ;  the  difference 
between  two  consecutive  states  when  both  x  and  y  v#ry  is 
called  the  total  differential,  and  is  denoted  by  the  ordinary 
symbol,  dz  without  a  subscript  letter.  Let  us  assume  the 
figure  and  notation  of  Art.  23.  We  have,  from  the  defini- 
tion, 

z'  =  z  +  (dz)x  ;  whence,  by  differentiation, 


=  (*),+  (<*'*),,,,  .....  (2) 
The  symbol  (dzz)x  y  indicates  the  result  obtained  by  dif- 
ferentiating z  as  though  x  were  the  only  variable,  and  then 


IMPLICIT    FUNCTIONS.  47 

differentiating  that  result  as  though  y  were  the  only  varia- 
ble, the  order  of  the  subscript  letters  indicating  the  order 
of  differentiation.  From  what  precedes,  we  also  have, 

z'"  =  z'  +  (dz')y; 

substituting  for  z'  and  (dz')  ,  their  values  taken  from  (2), 

we  have 


transposing  z  to  the  first  member,  replacing  z'"  —  z  by  its 
value  dz,  and  neglecting  the  part  (d*z)x    ,  because  it  is  an 

infinitesimal  of  the  second  order,  we  have,  finally, 


(3) 


Hence,  the  total  differential  is  equal  to  the  sum  of  the  par- 
tial differentials. 

EXAMPLES. 

1.  z  =  ax*  +  by3. 

(dz)x  —  2axdx-,  (dz)y  =  3by2dy  .*.  dz  =  Zaxdx  +  3by*dy. 

2.  z  =  axzy5.  Ans.  dz  =  2ay3xdx  +  3ax2yzdy. 

3.  z  =  x&.  Ans.  dz  —  yxy~1dx  +  xylxdy. 

Notation  employed  to  designate  Partial  Differential 
Coefficients. 

25.  From  the  nature  of  the  case,  there  can  be  no  such 
thing  as  a  differential  coefficient  of  a  function  of  two  va- 
riables; but  the  quotient  of  a  partial  differential  by  the 


48  DIFFERENTIAL   CALCULUS. 

differential  of  the  corresponding  variable,  is  called  a  par- 
tial differential  coefficient  The  form  of  the  symbol  indi- 
cates the  variable  with  reference  to  which  the  function  has 
been  differentiated,  and  no  subscript  letter  is  required. 
Thus,  in  Example  2,  Art.  24,  we  have, 

|?  =  2aij*x,  and  ^  =  3ax*y*. 

A  similar  notation  is  employed  for  partial  differential 
coefficients  of  a  higher  order,  as  will  be  seen  in  the  follow- 
ing article. 

Successive   Differentiation   of   Functions   of    two   Variables. 

26.  The  second  differential  of  a  function  of  two  varia- 
bles is  found  by  differentiating  the  first  differential  by  the 
rules  already  given.  The  third  differential  comes  from  the 
second  in  the  same  manner  that  the  second  comes  from 
the  first,  and  so  on.  In  finding  the  higher  partial  differen- 
tials, the  result  obtained  by  differentiating  the  function 
first  with  respect  to  x,  and  that  result  with  respect  to  y,  is 
the  same  as  though  we  had  differentiated  the  function  first 
with  respect  to  y,  and  the  result  with  respect  to  x.  For, 
from  Art.  24,  we  have, 

«"'  =  z'  +  (dz')y  =  z  +  (dz)x  +  (dz}y  +  (d»t)^ 
And  in  like  manner,  we  have, 

*'"  =  *"  +  (M\  =  z  +  (dz)y  +  (dz)x  +  (<*»*),,  . 
Equating  these  values  of  z'",  and  reducing,  we  and, 


IMPLICIT  FUNCTIONS.  49 

which  was  to  be  shown.    From  equation  (4),  by  an  exten- 
sion of  the  notation  in  Art.  25,  we  have, 


j(dz\        7(^\ 

d*z  _    d*z    Qr     \dx)  =  C\dy) 
dxdy       dydx  dy  dx 

From  what  precedes,  we  have, 


\  J 


•*•  **  =  (&*)*  .  +  (<***)*       +  (**)         +  (**) 


,  y  ,,  y  ,,  , 


EXAMPLES. 

1.  Given,  z  =  z*y*9  to  find  dzz. 
x 


Given,  z  =  y3^,  to  find 


50  DIFFERENTIAL   CALCULUS. 


Extension  to  three  or  more  Variables 

27.  If  we  have  a  function  of  three  or  more  independent 
variables,  we  may  find   its   differential  by  differentiating 
separately  with  respect  to  each  variable,  and  taking  the 
sum  of  the  partial  differentials  thus  obtained.     The  second 
and  higher  differentials  are  found  in  an  entirely  analogous 
manner. 

Definition  of  Explicit  and  Implicit  Functions. 

28.  An  explicit  function,  is  one  in  which  the  value  of 
the  function  is  directly  expressed  in  terms  of  the  variable. 

Thus,  y  =  \/2rx  —  x2,  is  an  explicit  function.  An  implicit 
function,  is  one  in  which  the  value  of  the  function  ife 
not  directly  given  in  terms  of  the  variable.  Thus,  in  the 
equation, 

ayz  +  bxy  +  ex2  +  d  =  0 ; 

y  is  an  implicit  function  of  x.  Implicit  functions  are 
generally  connected  with  their  variables  by  one  or  more 
equations.  When  these  equations  are  solved  the  implicit 
function  becomes  explicit. 

Differentials  of  Implicit  Functions. 

29.  The   differential   of  an   implicit   function   may  be 
found  without  first  finding  the  function  itself.     For,  if 
we  differentiate  both   members  of  the  first  equation,  we 
shall  thus  find  a  new  equation,  which,  with  the  given  one, 
will  enable  us  to  find  either  the  differential,  or  the  differ- 
ential coefficient  of  the  function.     For  example,  suppose 
we  have  ihe  equation, 

I  *  -h  2xy  4-  a"  -  a9  •=  0  .  .  .     .  (1) 


IMPLICIT   FUNCTIONS.  51 

to  find  the  differential  coefficient  of  y.     Differentiating 
(1),  and  dividing  by  2,  we  have, 

ydy  +  xdy  +  ydx  +  xdx  =  0 (2) 

Finding  the  value  of  -¥-.  we  have, 
dx 


4.— i 

dx 
Again,  let  us  have  the  relation, 

xy  =  m (1) 

to  find  the  value  of  C-j-.     Differentiating  (1),  we 

xdy  -f  ydx  =  0 (2) 

Whence, 

iH=-l <3> 

But  from  (1),  we  have, 

~x  ~  X*' 
Substituting  in  (3),  we  find, 

dv  m 


- 
dx          x* 


(4) 
^  ' 


EXAMPLES. 


Find  the  first  aau  second  differential  coefficients  of  y 
in  the  following  implicit  functions: 


52  DIFFERENTIAL   CALCULUS. 


We  have, 


also, 

+  3y*d*y-3d*y=0, 


y 


_ 
'  ~dx*  ~  ly*  '  dx*  ~  9  *  (l- 

2.  ?2  -  %mX      +  a;2  _  ^  =  Q. 


—  a;       , 

and  --    = 


.  —  -  --  ,         -5-        -  - 

dx      y  —  mx          dx9        (y  —  mx) 


PART  II. 

APPLICATIONS  OF  THE  DIFFEKENTIAL 
CALCULUS. 


I.  TANGENTS  AND  ASYMPTOTES. 

Geometrical  Representation  of  the  First  Differential 
Coefficient. 

30.  It  has  been  shown  (Art.  3),  that  any  function  of 
one  variable  maybe  represented  by  the  ordinate  of  a  curve, 
of  which  that  variable  is  the  cor- 
responding abscissa.  We  have  also 
seen  (Art.  5),  that  an  element  of 
the  curve,  PQ,  does  not  differ  from 
a  straight  line.  This  line,  prolonged 
toward  T,  is  tangent  to  the  curve 
at  P.  The  angle  that  the  tangent 
makes  with  the  axis  of  X  is  equal 
to  RPQ;  denoting  it  by  6,  and 
remembering  that  the  tangent  of  the  angle-  ;it  the  base 
of  a  right  angled  triangle  is  equal  to  the  piTjH-ndicular 
divided  by  the  base,  we  have, 


= 

dx 

The  tangent  of  d  is  taken  as  the  measure  of  tlu*  slope, 
not  only  of  the  tangent,  but  also  of  the  curve  at  the  point 
P.  Hence,  the  slope  of  a  curve,  at  any  point,  in 


54  DIFFERENTIAL   CALCULUS. 

by  the  first  differential  coefficient  of  the  ordinate  at  that 
point. 

The  slope  is  positive  or  negative,  that  is,  the  curve  slopes 
upward  or  downward,  according  as  the  first  differential 
coefficient  is  plus,  or  minus. 


Applications. 

31.  The  principle  just  demonstrated  enables  us  to  find 
the  point  of  a  curve,  at  which  the  tangent  makes  a  givm 
angle  with  the  axis  of  x. 

Thus,  let  it  be  required  to  find  the  point  on  a  given 
parabola  at  which  the  tangent  makes  an  angle  of  45°  with 
the  axis.  Assume  the  equation  of  the  parabola, 

dii      p 

.'.  -r  =  -• 
dx      y 

Placing  this  equal  to  1,  we  have, 

-  =  1,  or  y  =  p. 

y 

But  y  —  p,  is  the  ordinate  through  the  focus.  Hence,  the 
required  point  is  at  the  upper  extremity  of  the  ordinate 
through  the  focus. 

Again,  let  it  be  required  to  find  the  point  at  which  a  tan- 
gent to  the  ellipse  is  parallel  to  the  axis  of  x.  Assume  the 
equation, 

>    dy  _   _  l*x_ 
'*  Tx  ~   ~  a*y 

Placing  this  equal  to  0,  we  have, 

•-'&* 


TANGENTS    AND   ASYMPTOTES.  55 

which  can  only  be  satisfied  by  making  x  —  0;  this,  in  the 
equation  of  the  curve,  gives  y  =  ±  b.  Hence,  the  tangent 
at  either  vertex  of  the  conjugate  axis  fulfills  the  given 
condition. 

Again,  to  find  where  the  tangent  to  an  hyperbola  is  per- 
pendicular to  the  axis;  assume  the  equation  of  the  curve, 


whence,  by  the  rule, 

d-tf     lzx 

-y-  =-r-  =  co  ;    .*.  y  =  0,  x  =  ±  a. 

dx     azy 

Hence,  the  tangent  at  either  extremity  of  the  transverse 
axis,  fulfills  the  given  condition. 


Equations  of  the   Tangent  and  Normal. 

32.  Let  P  be  a  point  of  the  curve  whose  co-ordinates  are 
x",  y"  ;  then  will  the  equation  of  a  straight  line  through 
it,  be  of  the  form, 

y  -  y"  =  a(x  -  x")  .  .     .  .  (1) 

in  which  a  is  the  slope.     If  we  make  a  equal  the  tangent 
of  RPQ,  the  line  will  be  tangent  to  the  curve.     But  the 

tangent  of  RPQ  is  -r-fr\  hence,  we  have,  for  the  equation 
of  a  tangent  line, 


If  we  make  a  equal  to  minus  the  reciprocal  of  the  tan- 
gent of  RPQ,  the  line  will  be  perpendicular  to  the  tangent, 


56  DIFFERENTIAL   CALCULUS. 

that  is,  it  will,  be  normal  to  the  curve;  hence,  we  have,  for 
the  equation  of  a  normal  line, 


(3) 


Iii  these  equations  x  and  y  are  general  co-ordinates  of 
the  lines,  and  x",  y"  are  the  co-ordinates  of  the  point  of 
contact. 

Let  it  be  required  to  find  the  equation  of  a  tangent  to 
an  ellipse.  We  found,  in  the  last  article,  the  value  of 

-j-  =  --  j-  ;  substituting  for  x  and  y  the  particular  values 
u"  and  y",  and  putting  the  result  in  (2),  we  find, 

l>*z"  ',          ,,x 

y~y  =  ~~^/f(x~x)> 

whence,  by  reduction, 


Making  a  =  I,  in  this  equation,  we  have, 


yy"  +  xx"  =  a2, 


which  is  the  equation  of  a  tangent  to  a  circle. 
To  find  the  equation  of  a  normal  to  an  ellipse  we  substi- 

tute the  value  of  yr»  in  equation  (3).  which  gives, 


Mak  ng  a  —  I,  we  have, 

x 
which  is  the  equation  of  a  normal  to  a  circle. 


TANGENTS   AND  ASYMPTOTES.  57 

If,  in  equation  (2),  we  make  y  —  0,  the  coi  responding 
value  of  x  —  x"  will  express  the  length  of  the  subtangent, 
counted  from  the  foot  of  the  ordinate  through  the  point 
of  contact;  denoting  this  by  S.T.,  we  have, 


,T=-y"  .(6) 

9  dy" 

If,  in  equation  (3),  we  make  y  =  0,  the  corresponding 
value  of  x  —  x"  will  express  the  length  of  the  subnormal, 
counted  from  the  foot  of  the  ordinate  through  the  point 
of  contact;  denoting  this  by  8.N.,  we  have, 


(7) 


•Substituting  in  (6)  and  (7)  the  values  of  -y^7  taken  from 

Cv\b 

the  equation  of  the  ellipse,  we  have, 


Asymptotes. 

33.  An  asymptote  to  a  curve,  is  a  line  that  continually 
approaches  the  curve  and  becomes  tangent  to  it  at  an 
infinite  distance.  The  characteristics  of  an  asymptote 
are,  that  it  is  tangent  to  the  curve  at  a  point  that  is  infi- 
nitely distant,  and  that  it  cuts  one  or  both  axes  at  a  finite 
distance  from  the  origin. 

To  ascertain  whether  a  curve  has  an  asymptote,  assume 
the  equation  of  the  tangent  (Art.  32),  and  in  it  make 
x,  and  y,  successively  equal  to  0. 

3* 


58 


DIFFERENTIAL   CALCULUS. 


A 

Fig.  4. 


Jn  this  manner  we  find, 
4J)  =  y"  -  *"3L    and 


dy" 

We    then  find  —-77  from  the  _ 
dx 

equation  of  the  curve,  substi- 
tute it  in  (I)  and  make  the  hypothesis  that  places  the 
point  (x",  y'1)  at  an  infinite  distance.  If  this  supposition 
make  either  AD  or  A  T  finite,  the  curve  has  an  asymptote, 
otherwise  not. 

Let  it  be  required  to  find  whether  the  hyperbola  has  an 

dn"       bz  c" 
asymptote.     AYe  have  found  (Art.  31),  —-„  =  -r-Jfl  which 

J  '   dx        a2 y 

in  (1)  gives, 


Whence,  by  reduction, 

ffi 
~-^—,  nu&AT  = 


(3) 


or, 


y 


The  only  hypothesis  that  places  a  point  of  the  hyperbola 
at  an  infinite  distance  is  y"  —  ±  GO  ,  and  x"  =  ±  GO  ;  both 
these  sets  of  values,  ir.  (4),  make  AD  and  A  T  equal  to  0. 
Hence,  the  hyperbola  has  two  asymptotes,  both  passing 
through  the  centre. 


TANGENTS  AND  ASYMPTOTES.  59 

To  find  whether  the  parabola  has  an  asymptote,  make 
'  =       in  (1)'  whence> 


AD  =  y"  ~]~r=          and  AT  =  -x". 

\J  \J 

If  we  make  y"  —  GO  ,  and  x"  =  oo  ,  to  put  the  point  of 
contact  at  an  infinite  distance,  we  find  both  AD  and  AT 
equal  to  oo  ,  which  shows  that  the  parabola  has  no  asymp- 
tote. 

Order    of    Contact. 

34.  Two  lines  have  a  contact  of  the  first  order,  when 
they  have  two  consecutive  points  in  common.  The  first  of 
these  is  the  point  of  contact  ;  this  point  is  common  to  both 
lines,  and,  as  we  have  just  seen,  the  first  differential  coef- 
ficients of  the  ordinates  of  the  two  lines  at  this  -point  are 
equal.  Conversely,  if  two  lines  have  a  common  point,  and 
if  the  first  differential  coefficients  of  the  ordinates  of  the 
lines  at  that  point  are  equal,  the  lines  have  a  contact  of  the 
first  order. 

Two  lines  have  a  contact  of  the  second  order,  when  they 
have  three  consecutive  points  in  common.  The  first  of 
the  three  is  the  point  of  contact,  and  because  the  lines  have 
three  consecutives  ordinates  common,  the  first  and  second 
differential  coefficients  of  their  ordinates  at  this  point  are 
equal.  Conversely,  if  two  lines  have  a  common  point,  and 
the  first  and  second  differential  coefficients  of  their  ordi- 
nates at  that  point  equal,  the  lines  have  a  contact  of  the 
second  order. 

In  like  manner  it  may  be  shown,  if  two  lines  have  a 
point  in  common,  and  n  successive  differential  coefficients 
of  their  ordinates  at  that  point  equal,  that  the  lines  have 


60  DIFFERENTIAL   CALCULUS. 

a  contact  of  the  wth  order,  n  being  any  positive  whole 
number. 

If  a  line  oe  applied  so  as  to  cut  a  given  line  in  an  even 
number  of  points,  it  will,  just  before  the  first,  and  just 
after  the  last,  lie  on  the  same  side  of  the  given  line ;  and 
this  is  true  when  the  points  of  secancy  are  consecutive,  in 
which  case  the  lines  have  a  contact  of  an  odd  order. 
Hence,  if  two  lines  have  a  contact  of  an  odd  order,  they  do 
not  intersect  at  the  point  of  contact.  If  the  applied  line  cut 
the  given  line  in  an  odd  number  of  points,  it  will,  just 
before  the  first,  and  just  after  the  last,  lie  on  opposite  sides 
of  the  given  line;  and  this  is  also  true  when  the  points  of 
secancy  are  consecutive ;  hence,  if  two  lines  have  a  contact 
of  an  even  order,  they  intersect  at  the  point  of  contact. 

If  the  applied  line  be  straight,  and  the  contact  of  the 
nth  order,  the  given  curve  will  have  n  consecutive  values 
of  dy,  equal  to  each  other.  If  each  of  these  be  taken  from 
the  next  in  order,  the  differences  will  be  0,  that  is,  the 
curve  will  have  n  —  1  consecutive  values  of  d2y  equal  to 
0.  In  like  manner,  we  may  show  that  it  has  n  —  2  consec- 
utive values  of  d*y  equal  to  0,  and  so  on,  to  the  (n  4-  l)lh 
different  al  of  y,  which  will  not  be  0,  but  will  be  plus  or 
minus,  according  as  the  (n  4-  l)th  value  of  dy,  counting 
from  the  point  of  contact,  is  greater  or  less  than  the  nth 
value  of  dy.  Conversely,  if  the  (n  +  l)th  differential  of  y 
is  plus,  the  (n  4-  l)th  value  of  dy  is  greater  than  the  wth; 
if  minus  the  (n  4-  l)tb  value  of  dy  is  less  than  the  nih. 

II.   OURVATUKE. 
Direction  of  Curvature. 

35.  Let   KL   be   a   curve,   whose   concavity  is   turned 
downward,   that   is,   in   the   direction   of   negative   ordi- 


CURVATURE. 


Cl 


nates  ;  let  AP,  BQ,  and  CQ'  be  consecutive  ordinates,  Alt 
and  BO  being  equal  to  dx ;  prolong  PQ  toward  T.  Then 
will  RQ,  and  R  Q',  be  consecutive  values  of  dy,  and  con- 
sequently their  difference,  R'Q'  —  RQ,  will  be  the  value  of 
d*y  at  P.  The  right  angled  triangles,  PRQ,  and  QR'S, 
have  their  bases,  and  the  angles  at  their  bases,  equal ;  hence, 
their  altitudes,  RQ, 
and  R  8,  are  equal ; 
but  R'Q'  is  less  than 
R'S,  because  Q'  is 
nearer  —  QO  than  S ; 
hence,  R'Q'  is  less 
than  RQ,  and  con- 
sequently the  value 
of  d*y  is  negative. 
Conversely,  if  d~i/ 
is  negative,  R'Q'  is 
less  than  H'S,  and  the  curve  in  passing  from  P  is  concave 
doivmvard.  In  like  manner,  it  may  be  shown  that  the 
curve  is  concave  upward  at  P,  when  dzy  is  positive.  Be- 
cause the  sign  of  the  second  differential  coefficient  is  the 
same  as  that  of  the  second  differential,  we  have  the  follow- 
ing rule  for  determining  the  direction  of  curvature,  in 
passing  from  any  point  toward  the  right, 

Find  the  second  differential  coefficient  of  the  ordinate  at- 
the  point ;  if  this  is  negative,  the  curve  is  concave  down- 
ward, if  positive,  it  is  concave  upward. 

We  may  regard  y  as  the  independent  variable,  and  find 
the  second  differential  coefficient  of  x;  if  this  be  negative 
at  any  point,  the  concavity  is  turned  toward  the  left,  if 
positive,  toward  the  right. 

Example. — Let  it  be  required  to  determine  the  direction 
of  curvature  at  any  point  of  a  parabola. 


62  DIFFERENTIAL   CALCULUS. 

Assume  the  equation,  y2  =  2px  ;  whence,  by  differentia- 
tion and  reduction, 


p* 


p 


The  first  of  these  is  negative  for  positive  values  of  y,  and 
positive  for  negative  values  of  y  ;  hence,  the  part  of  the 
curve  above  the  axis  of  X  is  concave  downward,  and  the 
part  below  is  concave  upward.  The  second  is  positive  for 
all  values  of  x  andy;  hence,  the  curve  is  everywhere  con- 
cave toward  the  right. 

If  the  tangent  at  P  have  a  contact  of  the  nih  order,  n 
jeing  greater  than  1,  the  second  differential  coefficient  of 
y  at  that  point  will  be  0  (Art.  34).  In  this  case  it  may 
be  shown,  as  above,  that  the  curve  bends  downward  in 
passing  toward  the  right,  when  the  (n  +  l)th  value  of  dy, 
counting  from  P,  is  less  than  the  nth  value  of  dy,  and  up- 
ward when  the  (n  +  l)th  value  of  dy  is  greater  than  the  nth. 
The  former  condition  is  satisfied  when  the  (n  +  1)"'  differ- 
ential coefficient  of?/  is  negative,  and  the  latter  when  it  is 
positive  (Art.  34). 

Amount  of  Curvature. 

36.  The  change  of  direction  of  a  curve  in  passing  from 
an  element  to  the  one  next  in  order,  is  the  angle  between 
the  second  element  and  the  prolongation  of  the  first.  This 
angle  is  called  the  angle  of  contingence,  it  is  negative  when 
the  curve  bends  downward,  and  positive  when  it  bends 
upward.  The  total  change  of  direction  in  passing  over  any 
arc  is  equal  to  the  algebraic  sum  of  the  angles  of  contin- 
gence  at  every  point  of  that  arc. 

The  curvature  of  a  curve  is  the  rate  at  which  it  changes 
direction.  When  the  curvature  is  uniform,  as  in  the 


CURVATURE. 


63 


circle,  it  is  measured  by  the  total  change  of  direction  in 
any  arc.  divided  by  the  length  of  that  arc.  In  any  curve 
whatever  the  curvature  may  be  regarded  as  uniform  for 
an  infinitely  small  arc;  hence,  the  curvature  at  any  point 
of  a  curve,  is  equal  to  the  angle  of  contingence  at  that 
point,  divided  by  the  corresponding  element  of  the  curve. 
Denoting  the  angle  of  contingence  by  dd,  the  correspond- 
ing element  by  ds,  and  the  curvature  by  c,  we  have, 


Oscillatory  Circle. 

37.  An  oscillatory  circle,  to  a  given  curve,  is  a  circle  that 
has  three  consecutive  points  in  common  with  that  curve. 
Thus,  if  the  circle  whose  centre  is  C,  pass  through  the 
three  consecutive  points  P,  Q,  and  R,  of  the  curve,  KQ,  it 
is  osculatory  to  that  curve  at  P. 

The  circle  and  curve  have  two  consecutive  tangents,  PT 
and  QT,  and 
two  consecutive 
normals,  P  C  and 
QC,  in  common; 
they  have  also 
the  angle  of  con- 
tingence at  Q,  in 
common ;  hence, 
they  have  the 
same  curvature 
at  the  point  P. 
Because  P  C  and 
QC  are  perpen-  Fie-  6- 

dicular  to  PT,  anl  QT,  their  included  angle,  is  equal  to 


64  DIFFERENTIAL   CALCULUS. 

the  angle  of  contingency  dd;  hence,  PQ  =  CP  X  d\  or 
denoting  PQ  by  ds  and  CP  by  7?,  we  have,  ds  =  Held,  or, 


--- 

cfe     5 


This  equation  ^shows  that  the  curvature  of  a  curve  at 
any  point  is  equal  to  the  reciprocal  of  the  radius  of  the 
oscillatory  circle  at  that  point;  for  this  reason,  the  radius 
of  the  oscillatory  circle  is  called  the  radius  of  curvature. 

Radius  of  Curvature. 

38.  If  we  denote  the  inclination  of  the  tangent,  FT,  to 
the  axis  of  X  by  6,  the  angle  of  contingence,  TQT,  equal  to 

PCQ,  will  be  equal  tod&;  but  we  know  that  4  =  tan"1^; 

dx 
differentiating,  we  have, 

fg 

j   _       dx  dxdzy 

~  i    51  =  dy*  +  'dx*  ' 

*  dx* 

Finding  the  value  of  R  from  equation  (1),  Art.  37,  ana 
replacing  ds  by  its  value,  from  Art.  5,  we  have, 


_ 

~dxd*y 

Dividing  both  terms  of  the  second  member  by  c7.c3,  and 
denoting  the  first  differential  coefficient  of  y  by  q',  and  its 
second  differential  coefficient  by  q",  we  have, 


(2) 


CURVATURE.  65 

When  the  curve  bends  downward,  dd  is  negative,  and 
consequently  R  is  negative;  when  the  curve  bends  upward, 
both  arepositive. 

It  is  sometimes  convenient  to  regard  R,  or  some  other 
quantity  on  which  R  depends,  as  the  independent  variable. 
In  this  case  both  dy  and  dx  are  variable,  and  we  have, 

7        dxd9i/  —  dydzx 
dx*  +  rfy»  ~; 

and  this  in  (1),  Art.  37,  gives, 


_ 

dxd*y-dyd*x  '  '  (  } 

Example.  —  Required   the  value  of  R  for  the  parabola. 
Assume  the  equation,  y2  =  2px. 

By  differentiation  and  reduction,  we  have, 

dy  p  d2?/  p* 

-r  =  <1  =  -9  and  -^  =  q"  =  —  J—  • 
dx  y  dx2  ys' 

substituting  the.e  in  (2),  we  have, 


2J  P 

The  value  of  R  is  least  possible  when  y  =  0,  increases 
as  y  increases,  and  becomes  oo  when  y  =  QO  .  Hence,  the 
curvature  of  the  parabola  is  greatest  at  the  vertex,  and 
diminishes  as  the  curve  recedes  from  the  vertex. 


Co-ordinates  of  the  Centre  of  the  Osculatory  Circle. 

39.  In  the  right  angled  triangle,  PHC,  PH  is  perpen- 
dicular to  ED,  and  PC  to  EP ;  hence,  the  angle,  HPC,  is 


(16  DIFFERENTIAL   CALCULUS. 

equal  to  6.  Denoting  the  abscissa,  AC,  by  a,  and  the  ordi- 
nate,  DC,  by  /?,  we  have, 

a  =  AH+  HO,  and  £  =  BP  -  HP  .....  (1) 

All  and  BP  are  the  co-ordinates  of  P,  denoted  by  x  and 
y;  IIP  is  equal  to  PCcosJ,  and  HC  to  PCsind;  substi- 
tuting for  PC,  or  7?,  its  value  found  in  tlu  last  article,  re- 
membering that  it  is  negative  in  the  case  under  considera- 
tion. (Art.  38),  replacing  sin£,  and  cos4,  by  their  values 
taken  from  Art.  5,  and  reducing,  we  have, 


dx*  +  dy*      du 

-  --   —         ~-  =  x  — 


\  , 
}q 
J1 

' 


. 

dzy  dx 

'  ft) 


Locus  of  the  Centre  of  Curvature. 

40.  A  line  drawn  through  the  centres  of  the  oscillatory 
circles  at  every  point  of  a  curve  is  called  the  evolnte  of  that 
curve,  and  the  given  curve,  with  respect  to  its  evolute,  is 
called  an  involute.  If  we  consider  any  radius  of  curvature 
of  a  curve,  we  see  that  it  intersects  the  one  that  precedes 
and  the  one  that  follows  it,  and  the  distance  between  the 
points  of  intersection  is  an  element  of  the  evolute ;  hence, 
the  radius  is  normal  to  the  involute,  and  tangent  to  the 
evolute.  We  also  see  that  the  difference  between  two  con- 
secutive radii  of  curvature  is  equal  to  the  corresponding 
element  of  the  evolute;  hence,  the  difference  between  any 
two  radii  of  curvature  of  a  given  curve  is  equal  to  the  cor- 
responding arc  of  the  evolute. 

The  evolute  of  a  curve  may  be  constructed  by  drawing 
normals  to  the  curve,  and  then  drawing  *.i  curve  tangent 


CUKVATURE.  6? 

to  them  all:  the  closer  the  normals,  the  more  accurate  will 
be  the  construction.  An  involute  may  be  constructed  by 
wrapping  a  thread  round  the  evolute,  holding  it  tense, 
and  then  unwrapping  it.  Every  point  of  the  thread  de- 
scribes a  curve,  which  is  an  involute  of  the  given  evolute. 
Practically,  the  evolute  is  cut  out  of  a  board,  or  other  solid 
material.  In  this  manner  the  outlines  of  the  teeth  of 
wheels  are  sometimes  marked  out,  the  circle  being  taken 
as  an  evolute. 

Equation  of  the  Evolute  of  a  Curve. 

41.  The  equation  of  the  evolute  of  any  curve  may  be 
found  by  combining  the  equation  of  the  curve  with  form- 
ulas (2),  Art.  39,  and  eliminating  x  and  y.  The  method 
will  be  best  illustrated  \>y  an  example;  thus,  let  it  be  re- 
quired to  find  the  equation  of  the  evolute  of  the  common 
parabola. 


=  ^.and  n1'  —  —  ~ 

y  y/3 

substituting  these  in  (2),  Art.  39,  we  have,  after  reduction, 


We  have  already  found  a'  =    .and  n1'  —  —  ~  (Art.  38)  ; 

y  y/3 


a  —  x  +  -  --  —   =  3x  +  p,    •*.    X  —  -(a  —  p), 

6 


* 

substituting  the  values  of  x  and  y  in  the  equation  y2  = 
we  have, 


which  is  the  equation  required.     It  is  the  equation  oi  a 
semi-cubic  parabola. 


68  DIFFERENTIAL   CALCULUS. 

EXAMPLES. 

1.  Find  the  equation  of  the  evolute  of  an  ellipse 

Am.  (aa)*  +  («?)*  =  (a*  -  b 

2.  Find  the  equation  of  the  evolute  of  an  hyperbola. 

Ans.   (aaft  -  (bpft  =  (a*  +  b 

3.  Find  the  radius  of  curvature  of  an  ellipse. 


r/4/,4 

4.  Find  the  radius  of  curvature  of  an  equilateral  hyper- 
bola referred  to  its  asymptotes,  the  equation  being  xy  —  m. 

(m*  +  ?;4)* 
Ans.  o  =  s — -^-. 

»)  ,  1 1  »» .-i 


III.    SINGULAR  POINTS  OF  CURVES. 

Definition   of    a    Singular   Point. 

42.  A  SINGULAR  POINT  of  a  curve,  is  a  point  at  which  the 
curve  presents   some   peculiarity   not   common   to   other 
points.     The  most  remarkable  of  these  are,  points  of  in- 
flexion, cusps,  multiple  points,  and  conjugate  points. 

Points  of  Inflexion. 

43.  A  POINT  OF  INFLEXION  is  a  point  at  which  the  curva 
tu re  changes,  from  being  concave  downward  to  being  con- 
cave upward,  or  the  reverse. 

Inasmuch  as  the  direction  of  curvature  is  determined  by 
the  sign  of  the  second  differential  coefficient  of  the  ordi- 


felXGULAK    POINTS    OF   CURVES.  Oi* 

nate,  it  follows  that  this  sign  must  change  from  —  to  -f , 
or  from  +  to  — ,  in  passing  a  point  of  inflexion.  But  a 
quantity  can  only  change  sign  by  passing  through  0,  or  co' . 
Hence,  at  a  point  of  inflexion  the  second  differential  coef- 
ficient of  y  must  be  either  0,  or  co .  If  the  corresponding 
values  of  x  and  y  are  such,  that  for  values  immediately 
preceding  and  following  them,  the  second  differential  coef- 
ficient has  contrary  signs,  then  will  each  set  of  such  values 
correspond  to  a  point  of  inflexion. 

EXAMPLES. 

1.  To  find  the  points  of  inflexion  on  the  curve  whose 
equation  is  y  =  b  +  (x  —  a)3. 

We  find,  -=-|  =  G(x  —  a) ;    this,  put   equal   to   0,  gives 
clx 

x  •==  a,  which,  in  the  equation  of  the  curve,  gives  y  =  b. 
When  x  <  a,  the  second  differential  coefficient  is  negative, 
and  when  x  >  a,  it  is  positive.  Hence,  the  point  whose 
co-ordinates  are  a  and  b,  is  a  point  of  inflexion. 

2.  To  find  the  points  of  inflexion  on   the  curve  whose 
equation  is  y  =.  b  -f-  TV  (x  —  a)5> 

d^u 
We  find;  3^  =  2  (x  —  <?)3;    this  placed  equal  to  0,  gives 

x  =  af  whence,  y  —  b.  When  x  <  a,  the  second  differen- 
tial coefficient  is  negative,  and  when  x  >  a,  it  is  positive. 
Hence,  the  point  whose  co-ordinates  are  a  and  b,  is  a  point 
of  inflexion. 

3.  Find  the  co-ordinates  of  the  points  of  inflexion  on 


the  curve  whose  equation  is  y  —  %r 


o  o  _ 

Ans.  x  —  7r,  and  y  —  ^r Vs> 


70 


DIFFERENTIAL   CALCULUS. 


•1.  Find  the  co-ordinates  of  the  points  of  inflexion  on 

X2(a2   —  x2} 

the  line  whose  equation  is  y  =  — - — ^ -. 

,1/7  5 

Ans.  x  =  ±  -a  y  t>,  and  y  =  — #. 
o  oo 


Cusps. 

44.  A  CUSP  is  a  point  at  which  two  branches  terminate, 
being  tangential  to  each  other.  There  are  two  species  of 
cusps;  the  ceratoid,  named  from  its  resemblance  to  the 
horns  of  an  animal,  and  the  ramphoid,  named  from  its 
resemblance  to  the  beak  of  a  bird.  The  first  is  shown  at 
E,  and  the  second  at  A. 

The  method  of  determining  the  position  and  nature  of  a 
cusp  point  will  be  best  shown  by  examples. 

1.  Let  us  take  the  curve  whose  equation  isy=b±  (x—  a)i' 
From  this  we  find,  * 


du      .        ,  d*y 
For  x  =  a,  -+  =  0,  and  -=-f  = 
'  dx  dx* 


oo .      For  x  <  a,   both   are 


imaginary,  as  is  also  the  value  of  y.  For  x  >  a,  both  are 
real,  and  each  has  two  values,  one  plus  and  the  other 
minus.  Hence,  the  point  whose 
co-ordinates  are  a  and  b  is  a  cusp 
of  the  first  species. 

2.  Let  us  take  the  curve  whose 

equation  is  y  —  x2  ±  #f.  ' 
From  it  we  find, 


Fig.  7. 


dy            ,  5   }       ,d*y  ,  15    J 

^    —   y/y    _1_  _/»•*  Qfir]  ¥-.  —   9  -+-  T  6 

r^  —  /CX   m  ~  J,    ,  ctllU.  — : — r  — —  /v  -^-  ~r  Jj    • 

///*•  »>        '  /A>>2 


SINGULAR    POINTS   OF   CURVES. 


71 


From  the  given  equation  we  see  that  no  part  of  the 
curve  lies  to  the  left  of  the  axis  of  y,  and  that  there  are 
two  infinite  branches  to  the  right  of  that  axis. 

The  values  of  --  are  both  0  at  the  origin ;  and  at  that 

CLOu 

d2?/ 
point  both  values  of  -r-|  are  positive ;  hence,  the  origin  is 

a  cusp  of  the  second  species. 

A  discussion  of  the  above  equations  shows  that  the 
upper  branch  has  its  concavity  always  upward ;  the  second 

64 
branch  has  a  point  of  inflexion  for,  x  =  —  ;  its  slope  is  0 


225 


for  x  = 


__ 
25 


and  it  cuts  the  axis  of  x 


at  the  distance  1  from  the  origin  ;  from 
the  origin  to  the  point  of  inflexion, 
it  curves  upward;  after  that  point  it 
curves  downward.  The  shape  of  the 
curve  is  indicated  in  the  figure. 

For  the  ceratoid  the  values  of  the 
second  differential  coefficient  of  y  have 
contrary  signs,  for  the  ramphoid  they  have  the  same  sign. 


Fie?.  8. 


\ 


Multiple  Points. 

45.  A  MULTIPLE  POINT  is  a  point  where  two  or  more 
branches  of  a  curve  intersect,  or 
touch  each  other. 

If  the  branches  intersect,  -p  will 
dx 

have  as  many  values  at  that  point 
as  there  are  branches ;  if  they 
are  tangent,  these  values  will  be 
equal.  • 


Fig.  a. 


72  DIFFERENTIAL   CALCULUS. 

EXAMPLES. 

1.  Take  the  curve  whose  equation  is  yz  =  x2  —  x*. 

For  every  value  of  x  there  are  two  values  of  y  equal 
with  contrary  signs;  hence,  the  curve  is  symmetrically 
situated  with  respect  to  the  axis  of  x.  For  x  —  0,  we  have 
y  —  zfc  0  ;  hence,  both  branches  pass  through  the  origin. 
From  the  equation,  we  find, 

clli  -----  ±         !  2^2 

dx~y    '    y  'vT^^^VT^^' 

rl  /it 

For  x  =  0,  -j-  =  ±  1,  hence  the  origin  is  a  multiple  poinr, 

by   intersection.      The   curve    is    limited   in   both   direc- 
tions. 

2.  Take  the  curve  whose  equation  is  y  =  =fc  \/x*  +  x*. 

The  two  branches  are.  symmetrically  placed  with  respect 
to  the  axis  of  x,  both  pass  through  the  origin,  forming  a 
loop  on  the  left  and  extending  to  infinity  on  the  right. 
We  find,  from  the  equation  of  the  curve, 

dy  _         5x*  4-  4r3  _        5x2  +  ±x 
dx  ~  ~  " 


For  x  —  0,  we  have  -^  =  ±  0  ;  hence,  the  origin  is  a  mul- 
dx 

tiple  point,  the  branches  being  tangent  to  each  other  and 
to  the  axis  of  x. 

/~a*~—~xz 
3.  The  curve  whose  equation  is  y  =  ±  xA/  —  -  ^ 

has  a  multiple  point  at  the  origin,  and  is  composed  of  two 
pointed  loops,  one  on  the  right  and  the  other  on  the  left 
of  the  origin. 


SINGULAR    POINTS    OF    CURVES.  73 

4.  The  curve  whose  equation  is  y2  =  (x  —  2)  (x  —  3)2  v 
symmetrical  with  respect  to  the  axis  of  x  and  consists  of  a 
looped  branch  which  extends  from  x  =  2  to  x  =  3  ;  at  the 
latter  point  the  upper  part  passes  below  the  axis  of  x  and 
lower  part  passes  above  that  axis,  forming  a  multiple  point ; 
beyond  x  =  3  the  two  parts  continually  diverge,  extending 

tO  X  —  CO  . 

Conjugate  Points. 

46.  A  CONJUGATE,  or  ISOLATED  POINT,  is  a  point  whose 
co-ordinates  satisfy  the  equation  of  a  curve,  but  which  has 
no  consecutive  point.  Because  it  has  no  consecutive  point, 
the  value  of  the  first  differential  coefficient  of  y  at  it,  is 
imaginary. 

EXAMPLES. 

1.  Take  the  curve  whose  equation  is  y  =  ±  x  V%  —  a. 

In  this  case  x  —  0  and  y  =  0,  satisfy  the  equation,  but 
no  other  value  of  x  less  than  a  gives  a  point  of  the  curve. 
Furthermore,  we  have, 

dy_  _        1  3x*  -  2ax 
dx~        2  vz3  —  ax2' 

which   for  x  =  0  is  imaginary,  as  also  for  all  values  of 
x  <  a. 

2.  Take  the  curve  whose  equation  is 


y  =  ax2  ±       zl 

For  every  positive  value  of  x,  there  are  two  values  of  y, 
and  consequently  two  points,  except  when  cosx  =  1,  when 
the  two  points  reduce  to  one.  These  points  constitute  a 
series  of  loops  like  the  links  of  a  chain,  having  for  a  dia- 

4 


74  DIFFERENTIAL   CALCULUS. 

metral  curve  a  parabola  whose  equation  is  y  =  ax2.  For 
every  negative  value  of  x  the  values  of  y  are  imaginary, 
except  when  cosz  =  1 ;  in  these  cases  we  have  a  series  of 
isolated  points  situated  on  the  diametral  curve,  whose 
equation  is  y  —  ax2.  We  have,  by  differentiating  and 
reducing, 


dx 


which  is  imaginary  for  negative  values  of  x. 


.    IV.  MAXIMA  AND  MINIMA. 

Definitions  of  Maximum  and  Minimum. 

47.  A  function  of  one  variable  is  at  a  maximum  state, 
when  it  is  greater  than 
the  states  that  im- 
mediately precede  and 
follow  it;  it  is  at  a 
minim  urn  state,  when 
it  is  less  than  the  states 
that  immediately  pre- 
cede and  follow  it. 
Thus,  if  KL  be  the 
curve  of  the  function, 
BN  will  be  a  maxi- 
mum, because  it  is 
greater  than  AM  and 
(70,  and  EQ  will  be 
a  minimum,  because  it 
is  loss  than  DP  and 
FR. 


1 

[   S  *" 

M 

/ 

\0 

VI 

^^/ 

x  

Q 

K 

L, 

)              ABC              DBF 

Fig.  10. 

MAXIMA    AND    MINIMA.  75 

Analytical  Characteristics  of  a  Maximum   or  Minimum. 

48.  If  we  examine  the  figures  given  in  the  last  article, 
we  see  that  the  slope  of  the  curve  KL,  or  the  differential 
coefficient  of  the  function,  changes  from  +  to  —  in  pass- 
ing a  maximum,  and  from  —  to  +  in  passing  a  minimum. 
But  a  varying  quantity  can  only  change  sign  by  passing 
through  0,  or  oo ;  hence,  the  first  differential  coefficient 
of  the  function  must  be  either  0,  or  oo,  at  either  maximum 
or  minimum  state. 

If,  therefore,  we  find  the  first  differential  coefficient  of  the 
function,  and  set  it  equal  to  0  and  oo,  we  shall  have  two  equa- 
tions which  will  give  all  the  values  of  the  variable  that  belong 
to  either  a  maximum  or  minimum  state  of  the  function. 

They  may  also  give  other  values ;  hence  the  necessity 
of  testing  each  root  separately.  This"  may  be  done  by  the 
following  rule : 

Subtract  from,  and  add  to,  the  root  to  be  tested,  an  infi- 
nitely  small  quantity;  substitute  these  successively  in  the 
first  differential  coefficient  ;  if  the  first  result  is  plus  and 
the  second  minus,  the  root  corresponds  to  a  maximum  ;  if 
the  first  is  minus  and  the  second  plus,  it  corresponds  to  a 
minimum;  if  both  hare  the  same  sign,  it  corresponds  to 
neither  a  maximum  nor  a  minimum. 

The  maximum  or  minimum  value  may  be  found  by  sub- 
stituting the  corresponding  root  in  the  given  function. 

EXAMPLES. 

1.  Let  y-  3  +  (x-2)2- 

By  the  rule,  we  have,  ^  =  2(x  —  2)  =  0,     :.  x  —  2. 
Substituting  for  x  the  values,  2  —  dx,  and  2  +  dx,  we  find 


DIFFERENTIAL    CALCULUS. 


for  the  corresponding  values  of  the  differential  coefficient 
—  2dx,  and  +  2dx  ;  hence,  x  =  2,  corresponds  to  a  mini- 
mum, which  is  y  =  3, 

2.  Let  y  =  4  -  (x  -  3)f 
We  have, 


For  x  =  3  —  ofe,  and  3  -f  do;,  we  have  the  differential  coef- 
ficient equal  to 

_J2  ,  _     _2_ 

a  * 


hence,  x  —  3  corresponds  to  a  maximum,  y  =  4. 

3.  Let*/  =  3  +  2(z-l)3- 
We  have, 


Substituting  1  —  dx,  and  1  +  dx,  for  re,  in  the  first  differ- 
ential coefficient,  we  have  in  both  cases  a  positive  result; 
hence,  x  —  1  does  not  correspond  to  either  a  maximum,  or 
minimum. 

The  preceding  method  is  applicable  in  all  cases  ;  but, 
when  the  first  differential  coefficient  of  the  function  is  0, 
as  it  is  in  most  instances,  there  is  an  easier  process  for 
testing  the  roots.  In  this  case,  the  function  being  repre- 
sented by  the  ordinate  of  a  curve,  the  maximum  and 
minimum  states  correspond  to  points  at  which  the  tangent 
is  horizontal  ;  furthermore,  the  curve  lies  wholly  above,  or 
wholly  below,  the  tangent  at  the  point  of  contact,  that  is, 
the  tangent  does  not  cut  the  curve  at  that  point;  hence, 
the  contact  is  of  an  odd  order,  and  consequently  the  first 
of  tit  e  successive  differential  coefficients  of  the  function  that 


MAXIMA    AND    MINIMA.  77 

does  not  reduce  to  0  must  be  of  an  even  order  (Art.  34). 
If  this  is  negative,  the  curve  bends  downward  after  passing 
the  point  of  contact,  and  the  root  corresponds  to  a  maxi- 
mum, if  positive  the  curve  bends  upward,  and  the  root  cor- 
responds to  a  minimum  (Art.  35).  Hence  the  following 
practical  rule  for  finding  the  values  of  the  variables  that 
correspond  to  maxima  and  minima: 

Place  the  first  differential  coefficient  of  the  function  equal 
to  0,  and  solve  the  resulting  equation;  substitute  each  root 
in  the  successive  differential  coefficients  of  the  function, 
until  one  is  found  that  does  not  reduce  to  0  ;  if  this  is  of 
an  even  order  and  negative,  the  root  corresponds  to  a  maxi- 
mum, if  of  an  even  order  and  positive,  to  a  minimum;  but 
if  of  an  odd  order,  it  corresponds  to  neither  maximum  nor 
minimum. 

EXAMPLES. 

1.  Let  y  =  xz  —  3x  +  2. 
We  have, 


Also, 


-2-  •       -+2 

fa*  '  Uc»J| 


Hence,  x  =  -,  corresponds  to  a  minimum  state,  which  is, 


The  symbol  [Taj  3  is  use<*  to  denote  wliat  jr?  hi-c 

3 

when  the  variable  that  enters  it  is  made  equal  to  '-. 


78  DIFFERENTIAL   CALCULUS. 

2.  y  =  4-  (z-3)4. 


we  also  find, 

=-24. 


Hence,  [?/]3  —  4,  is  a  maximum. 

In  applying  the  above  rule,  a  positive  constant  factor 
may  be  suppressed  at  any  stage  of  the  process;  for  it  is 
obvious  that  such  suppression  can  in  no  way  alter  the 
value  of  the  roots  to  be  found,  or  the  signs  of  the  successive 
differential  coefficients. 

3.  Let  y  =  3z3  —  9z8  —  27z  +  30. 

Rejecting  the  factor,  +  3,  and  denoting  the  result  by  u,  we 
have, 

du  -      z  -<u  2      o        q\  — o- 

dx~ 

Rejecting  the  factor,  +  3,  and  denoting  the  resulting  value 
of  the  function  by  u'9  we  have, 

— —      =  4,  and     —r—-         =  —  4. 
Lax  J3  Lax     —i 

The  first  corresponds  to  a  minimum  and  the  second  to  a 
maximum.  Substituting  in  the  given  function,  we  have, 
for  the  maximum  and  minimum  values, 

y'  =  45,  and  y"  =  —  51. 
When  the   first   differential  coefficient  is  composed  of 


MAXIMA    AND    MINIMA.  79 

\**'tobk>  factors,  the  method  of  finding  the  corresponding 
vaiues  ol  the  second  differential  coefficient  may  be  simpli- 
fied as  follows:  let  us  have, 

'iR  -  p  x  o  . 

5=  l  x^> 

P  and  Q  being-  functions  of  x,  and  suppose  that  [P]a  =  0. 
3j  the  rule  for  differentiating,  we  have, 


, 

dx*  '      dx 


but  because  P  Decomes  0  when  x  =  a,  we  have, 


dxJa 

Hence,  find  }lie  differential  coefficient  of  the  factor  that 
reduces  to  0,  ?),  tiUiirty  it  by  the  other  factor,  and  substitute 
the  root  in  th0,  product.  The  result  is  the  same  as  that 
found  by  substituting  the  root  in  the  second  differential 
coefficient. 

4.  Let  #  =  (x  -  3)z(x  —  2). 
We  have, 

=  2(x  -  3)  (x  -  '2)  -H  (x  -  3)2  =  (x  -  3)(3a?  -  7)  =  0; 


By  the  principle  just  demonstrated, 


4 
Hence,   [y]    —  0,  is  a  minimum,  and   [«/].?  =  —  ,  is  a 


maxmum. 


SO  DIFFERENTIAL   CALCULUS. 

If  y  represents  any  function  of  x9  and    if  u  =  yn,  we 
have, 

^_  nyn-\dy 
dx  ~  ny        dx 


If  x  =  a  reduces  ~  to  0,  but  does  not  reduce  yn~*  to  U. 
dx 

we  have,  from  what  precedes, 


If  n  is  positive,  any  value  that  makes  y  a  maximum,  or 
minimum,  "also  makes  u  a  maximum,  or  minimum  ;  if  w 
is  negative,  any  value  that  makes  y  a  maximum,  or  »i{m- 
?w?/m,  makes  w  a  minimum,  or  maximum.  The  converse 

is  also  true,  if  we  except  the  values  that  make  yn~^  equal 
to  0.  Hence,  if  care  be  taken  to  reject  the  exceptional 
values,  we  may  throw  off  a  radical  sign  in  seeking  for  a 
maximum  or  minimum. 


5.  Let  y  = 

Throwing  off  the  radical  sign,  suppressing  the  fact   ', 
+  2«,  and  denoting  the  result  by  u,  we  have, 


The  value,  x  =  0,  makes  y  =  0,  and  is  to  be  rejected. 
The    value     x  =  -fa     makes     -^— 2     negative,    an  1    gives 

y  =  —  '-:=,&  maximum. 
3  A/3 


MAXIMA    AND   MINIMA.  81 

In  like  manner,  if  we  have  u  —  ly,  we  have. 

du       1      dy        ,  .„  rdy^\ 

-j-  =—  .  ~,  and  if    -r     ~  ®>  we  nave>  as  before, 

dx      y      dx  LdxJa 


rd2u-\     _  r  1       d*y~\ 
Ldx*\a~  Ly    '  dx2^' 


Hence,  we  infer,  as  before,  that  we  may,  with  proper  pre- 
caution, treat  the  logarithm  of  a  function  instead  of  the 
function  itself,  and  the  reverse. 

x2  —  3x  +  2  (x  —  i\(x  —  2) 

6.  Let  y  =  ^^~^,  or  y  =  ^^^j. 

Passing  to  logarithms,  and  denoting  ly  by  u,  we  have. 

u  =  l(x  -  1)  +  l(x  -2)-  l(x  +  1)  -  l(x  +  2). 
Whence, 


___ 
dx       x  -  1      x  —  2      x  +  1       x  +  2 

6(x*  -  2) 

=°;  A-^>: 


Both  values  of  x  make  y  negative.     The  first  makes 

dPu 

-T-J  negative,  and  the  second  makes  it  positive.     Hence  we 

have,  y  =  12\/2  —  17,  minimum,  and  y  =  —  12\/2  —  17, 
maximum. 


PROBLEMS   IN   MAXIMA    AND   MINIMA. 

1.  Divide  21  into  two  parts,  such  that  the  less  multiplied 
by  the  square  of  the  greater,  shall  be  a  maximum. 

Solution.  —  Let  x  he  the  greater,  and  21—  x  the  less.    We 
4* 


8*  DIFFERENTIAL   CALCULUS. 

have,  for  the  equation  of  the  problem,  y  —  (21  —  x)x2. 
By  the  rule  we  find  that  x  —  14  makes  y  a  maximum, 
Hence,  the  parts  are  14  and  7. 

2.  Find  a  cylinder  whose  total  surface  is  equal  to  S,  and 
whose  volume  is  a  maximum. 

Solution.  —  Denote  the  radius  of  the  baee  by  x,  the  alti- 
tude by  y,  and  the  volume  by  V.  From  the  geometrical 
relations  of  the  parts,  we  have, 


S  =  2«xz  +  2*x  X  y,   and  V  —  vx2  X  y. 
Combining,  we  have,  for  the  equation  of  the  problem, 


0  , 

V=  «xz  -  -        -  =  -  (Sx  - 

2ifx  2  ^ 

/~fi  /~S 

V  is  a  maximum  when  x  —  A  /  —  ;  whence,  y  =  2  A  /  ~, 

I/     6tf  I/     or 

or  y  —  2x.      That  is,  the  altitude  is   equal  to  twice  the 
radius  of  the  base. 

3.  Find  the  maximum  rectangle  that  can  be  inscribed 
in  an  acute-angled  triangle  whose  base  is  14  feet,  and  alti- 
tude 10  feet. 

Ans.  The  base  is  7  feet,  and  the  altitude  5  feet. 

4.  Find  the  maximum  triangle  that  can  be  constructed 
on  a  given  base,  and  having  a  given  perimeter. 

Solution.  —  Denote  the  base  by  5,  a  second  side  by  x,  the 
third  side  by  %p  —  b  —  x,  the  perimeter  by  2p,  and  the 
area  by  A.  From  the  formula  for  the  area  in  terms  of 
the  three  sides,  we  find  the  equation  of  the  problem, 


A  =  Vp(p  -  b)  (p  —  x)(b+x  -p). 

For  x  =  p  —  JZ>,  A  is  a  maximum.     Hence,  the  triangle  is 
isosceles. 


MAXIMA   AND    MINIMA.  83 

5.  To  find  the  maximum  isosceles  triangle  that  can  be 
inscribed  in  a  circle. 

Ans.  An  equilateral  triangle. 

6.  To  find  the  minimum  isosceles  triangle  that  can  be 
circumscribed  about  a  circle. 

Ans.  An  equilateral  triangle. 

7.  To  find  the  maximum  cone  that  can  be  cut  from  a 
sphere  whose  radius  is  r. 

Ans.  The  radius  of  the  base  is  —  ^  —  ,  and  the  altitude  is  -r. 

o  o 

8.  To  find  the  maximum  cylinder  that  can  be  cut  from 
a  cone  whose  altitude  is  h,  and  the  radius  of  whose  base 

is  r. 

2  1 

Ans.  The  radius  of  its  base  is  -r,  and  its  altitude  -Ji. 

o  o 

9.  To  find  the  altitude  of  the  maximum  cylinder  that 

can  be  cut  from  a  sphere  whose  radius  is  r. 

o 

Ans.  Altitude  = 


10.  To  find  the  maximum  rectangle  that  can  be  inscribed 
in  an  ellipse  whose  semi-axes  are  a  and  b. 

Ans.  The  base  is  a\/2,  and  the  altitude 


11.  To  find  the  maximum  segment  of  a  parabola  that 
can  be  cut  from  a  right  cone  whose  altitude  is  h,  and  the 

radius  of  whose  base  is  r. 

g     __ 
Ans.  The  axis  is  equal  to  -V^2  +  ?'*• 

12.  To  find  the  maximum  parabola  that  can  be  inscribed 
in  a  given  isosceles  triangle. 

q 

Ans.  The  axis  is  equal  to  -tbB  the  altitude. 


84  DIFFERENTIAL   CALCULUS. 

13.  To  find  the  maximum  cone  whose  surface  is  c\  a- 

stant,  and  equal  to  S. 

1  /S 
Ans.  The  radius  of  the  base  is  ^y  —  • 

14.  To  find  the  maximum  cylinder  that  can  be  cut  from 
&  given  oblate  spheroid,  whose  semi-axes  are  a  and  b. 

Ans.  The  radius  of  the  base  =  a\  ,5, 

o 

2 

and  the  altitude  = 


15.  From  the  corners  of  a  rectangle  whose  sides  are  a 
and  bf  four  squares  are  cut  and  the  edges  turned  up  to  form 
a  rectangular  box.  Required  the  side  of  each  square  when 
the  box  holds  a  maximum  quantity. 


a  +  b      1 


.  •  -  a 7o 

Ans.  — TT -zVa*  —  ab  +  b2, 

o  o 

16.  To  find  the  minimum  parabola  that  will  circum- 
scribe a  given  circle. 

Solution. — Let  x  and  y  be  the  co-ordinates  of  the  point 
of  contact,  a  and  b  the  terminal  co-ordinates,  2p  the  vary- 
ing parameter,  and  r  the  radius  of  the  circle.  For  any 
circumscribed  parabola,  we  have, 


a  =  x  +  p  +  r,  and  b  =  v2pa  =  v%p(x  +  p  +  r)  .....  (1) 

But  from  the  triar.gle,  whose  sides  are  the  radius,  the  sub- 
normal, and  the  ordinate  of  the  point  of  contact,  we  have, 


and  b  =p  +  r  .....  (2) 

It  will  be  shown  hereafter  that  the  area  of  a  parabola  is 
two-thirds  the  rectangle  having  the  same  base  and  altitude, 


MAXIMA    AND   MINIMA.  85 

Assuming  this  property,  and  denoting  the  area  by  A,  we 
have,  for  the  equation  of  the  problem, 


Applying  the  rule,  and  remembering  that  p   and  A    are 
variable,  we  find  that  A  is  a  minimum  when  %p  =  **. 

17.  Find  the  maximum  difference  between  the  sine  and 

versed-sine  of  a  varying  angle. 

Ans.  When  the  arc  is  45°. 

18.  Find   the   maximum   value   of  y    in  the  equation 

y  =  £-*>. 

Ans.  y'  —  e*. 

19.  Divide  the  number  36  into  two  factors  such  that  the 
sum  of  their  squares  shall  be  a  minimum. 

Ans.  Each  factor  is  equal  to  G. 

20.  Divide  a  number  m  into  such  a  number  of  equal 
parts  that  their  continued  product  shall  be  a  maximum. 

Tfl 

Solution.  —  Let  x  denote  the  number  of  parts,   -  -  one 

x 

part,  and  (     J  the  continued  product;  hence,  the  equation 
of  the  problem  is, 

fm\x  ,m 

y  =  (      1    ;   or,  ly  =  xl  —  =  xim  —  xlx. 
\x  /  x 


m  & 

.'.  x  =  —  ,  and  y  —  e 


86  DIFFERENTIAL   CALCULUS. 


m  m 

~e         e  .        ~e 

3    X  —  ',  .*•  y  ==  ^  , 

m         9 


is  a  maximum. 

.21.  What  value  of  x  will  make  the  expression , 

1  +  tana: 

&  maximum? 

Ans.  x  =  45°. 

22.  Find   the  fraction  that  exceeds  its   square  by  the 
greatest  possible  quantity. 

Ans.  \. 

23.  The  illuminating  power  of  a  ray  of  light  falling 
obliquely  on  a  plane  varies  inversely  as  the  square  of  the 
distance  from  the  source,  and  directly  as  the  sine  of  its 
inclination  to  the  plane. 

How  far  above  the  centre  of  a  horizontal  circle  must  a 
light  be  placed  that  the  illumination  of  the  circumference 

may  be  a  maximum  ?  r 

Ans.   H -. 

~V2 

Maxima  and  Minima  of  a  Function  of  Two  Variables, 
49.  A  function  of  two  variables  is  at  its  maximum  state 
when  it  is  greater,  and  at  its  minimum  state  when  it  is  less 
than  all  its  consecutive  states. 

If  two  parallel  sections  (Art.  23)  be  taken  through  a  maxi- 
mum or  minimum  ordinate  of  a  surface,  the  ordinate  will  be  a 
maximum  or  minimum  in  each  section,  and  in  nearly 
every  practical  case  the  reverse  will  hold  true.  The  ex- 
ceptional cases  will  be  those  in  which  the  maximum  or 
minimum  ordinate  corresponds  to  a  singular  point  of  the 
surface.  Omitting  these,  the  problem  is  reduced  to  find- 
Ag  an  ordinate  that  shall  be  a  maximum,  or  a  minimum 


MAXIMA    AXD    MINIMA.  87 

ir«  both   the  parallel  sections  through  it     This  requires 

that  —  and  -7-  be  simultaneously  equal  to  0.     The  equa- 
dy          dx 

tions  thus  obtained  will  determine  all  the  values  of  x  and 
y  that  correspond  to  maxima  or  minima  states,  and  each 
set  of  such  values  may  be  tested  in  the  manner  explained 
in  the  last  article,  observing  that  for  the  section  parallel 
to  the  plane  xz,  the  successive  differential  coefficients  of  z, 
are  found  by  supposing  y  constant,  and  for  the  section 
parallel  to  yz,  they  are  found  by  supposing  x  constant. 


EXAMPLES. 

1.  Let  z  =  xs  —  3xy  +  y*. 
We  have, 

^  =  3x*  -  3y  =  0,  and  -^  =  3y2  -  3x  =  0; 
ctzc  if 

.:  x  =  0,  y  =  0;  and  x  =  1,  y  —  1. 
Also, 

d*z       ,.     d*z       ,         .  d*z      a      d*z 
—  —  G%  ——  =  b  ;  and  -y-r  =  6y,  -=-=  =  6. 
dxz  dxz  dyz  dy3 

The  first  set  of  values  reduce  the  second  differential 
coefficients  to  0,  but  not  the  third  differential  coefficients  ; 
hence,  they  are  to  be  rejected.  The  second  set  of  values 
make  both  of  the  second  differential  coefficients  positive; 
hence,  they  correspond  to  a  minimum,  which  is  z  =  —  1. 

2.  Let  z  = 


Ill 
For,   a;  =  -,  y  =  -,   z  =        ,  a  maximum. 


3.  Let  z  =  Vp(p  —  x)(p  —  y}(x  -f  y  —  p). 
Dropping  the  radical  sign,  passing  to  logarithms,  and 
denoting  the  logarithm  of  z2  by  u,  we  have, 

u  =  l(p)  +  l(p  -x)  +  l(p  -  y}  +  l(x  +  y  -  p). 


88  DIFFERENTIAL  CALCULUS. 

Hence, 

du  1 


dx          p  —  x      x  +  y—p 
du_  1 

dy       p  —  y     x  -^-  y  —  p 

2  2 

x  —  3  p,  y  =  5  j9,  make  z  a  maximum. 
o  o 


V.  SINGULAR  VALUES  OF  FUNCTIONS. 

Definition  and  Method  of  Evaluation. 

50.  A  singular  value  of  a  function  is  one  that  appears 
under  an  indeterminate  form,  for  a  particular  value  of  the 

variable.     Thus,  the  expression  --  -  —  reduces  to  -,  for 

xz  0 

the  particular  value,  x  —  0,  whereas  its  real  value  is  -. 

Singular  values  that  take  the  above  form  of  indeter- 
mination,  may   often   be   detected,  and   their  real   value 

found  by  means  of  the  calculus.     Let  u  =  -,  in  which  2 

y 

and  y  are  functions  of  x  that  reduce  to  0  for  x  —  a.    Clear- 
ing of  fractions  and  differentiating,  we  have, 

udy  +  y  du  —  dz  ; 
if  we  make  x  —  a,  the  second  term  disappears,  and  we  have, 


If  both  dz  and  dy  reduce  to  0  for  x  =  a,  we  have,  in  like 
manner, 


and  so  on.     Hence,  the  following  rule: 


SINGULAR   VALUES   OF   FUNCTIONS.  89 

Divide  tlie  successive  differentials  of  the  numerator  ly  the 
corresponding  differentials  of  the  denominator;  substitute 
the  particular  value  of  the  variable  in  the  resulting  fraction, 
continuing  the  operation  till  one  is  reached  that  is  not  inde- 
terminate; the  value  thus  found  is  the  value  required. 

Thus,  in  the  example  above,  we  have, 

tx  —  sinari         rl  —  cosan     _  rsinari     _  rcosari     _  1 
S»      JO  ~~  L     3x*     Jo  .     L~6^  Jo  ~~  L~6~Jo  ""  6* 


EXAMPLES. 

1.  Find  the  value  of  |  -- z —      -J    .         Ans.  "4. 

%.  Find  the  value  of  [  4  _  ^   ~  ^       -^]   .     Ans.  g. 

p#3 (i%x OX^  -f-  Ct>^~\ 

3.  Find  the  value  of  '—= .      Ans.  0. 

L  x    —  a  -Jo, 

rl  —  xn~\ 

4.  Find  the  value  of    -— •  .  Ans.  n. 

\—  J.  —  X  J 1 

rex  —  e~x~\ 

5.  Find  the  value  of    ^7—-  ^-      .  Ans.  2. 

L/(J  +  x)  JO 

6.  Find  the  value  of  f"1  ~  cos_5]  .  Ans.  0. 

L  i?/  JO 


7.  Find  the  value  of 


8.  Find  the  value  of  jjrZj  •  An*- 


DIFFERENTIAL   CALCULUS. 


9.  Find  the  value  of  -  Ans.  - , 

l  -f  tanaJtf  4 


10.  Find  the  value  of  [—~-¥JjL  ^£~\  .  Ans.  2. 

L      -          *  -    z-*a 


Singular  values  that  appear  under  the  forms  0  X  <*>, 
—  ,  and  GO  —  oo  ,  can  be  reduced  to  the  form  discussed,  and 

then  treated  by  the  rule.     The  method  of  proceeding  in 
each  case  will  be  illustrated  by  an  example. 

11.  The  expression  (1  —  x)  tan  ~9  which  reduces  to  0  x  oo  , 

M 

1     _    % 

when  x  =  1,  can  be  placed  under  the  form  —      —  ,  which, 


2 
by  the  rule,  reduces  to  -  for  the  particular  value,  x  =  L 


f~r/ 

12.  The  expression,  tan-^-  ~  j—=  --  —  ,  which  reduces  to 
*        (x    ~~  1) 

—•  for  x  =  1,  can  be  placed  under  the  form  -        —  ,  and 


. 
this,  by  the  rule,  gives  for  x  =  1,  the  value  --  . 


13.  The   expression,  ztanz  —  -  sec.r,  which  reduces  to 

/c 

if  crsinz  —  i  r 

&  —  oo ,  for  the  value  x  =  -,  may  be  written  -          -=-, 

2  cosx 

which,  by  the  rule,  gives  for  x  =  -^,  the  value  —  1. 

Al 


ELEMENTS    OF    GEOMETRICAL    MAGNITUDES. 


91 


VI.  ELEMENTS  OF  GEOMETRICAL  MAGNITUDES. 

Differentials  of  Lines,  Surfaces,  and  Volumes. 

51.  Let  AP  and  BQ  be  two  consectVive  ordimites  of  the 
curve,  IfL,  whose  equation  is  y  =f(x)',  then  will  PQ  be 
the  differential  of  the  length  of  the 
curve,  and  APQB  will  be  the  dif- 
ferential of  the  area,  bounded  by 
the  curve,  the  axis  of  x,  and  any 
two  ordinates;  if  we  suppose  the 
figure  to  revolve  about  the  axis  of 
x,  the  curve  will  generate  a  surface 
of  revolution,  and  the  area  between 
the  curve  and  axis  will  generate  a 
volume  of  revolution ;  the  surface  generated  by  PQ,  is  the 
differential  of  the  surface  of  revolution,  and  the  volume 
generated  by  APBQ,  is  the  differential  of  the  volume  of 
revolution.  When  the  equation  of  KL  is  given,  the  values 
of  these  differentials  may  always  be  found  in  terms  of  x 
and  dx,  or  of  y  and  dy. 

1°.  Denote  PQ  by  dL  ;  we  shall  have,  as  in  Art.  5, 


Fig.  11. 


dL  = 

To  apply  this  formula,  we  differential  the  equation  ol 
the  cnrve,  combine  the  resulting  equation  with  that  of  the 
curve,  so  as  to  find  the  differential  of  one  variable  in  terms 
of  the  other  variable  and  its  differential,  and  substitute 
this  in  the  formula.  Thus,  let  it  be  required  to  find  the 
differential  of  the  arc  of  a  parabola.  Assuming  the  equa- 
tion of  tLc  parabola,  yz  =  Zpx,  we  find,  by  differentiation 
and  combination, 


DIFFEKENTIAL   CALCULUS. 


dy  =  faA/jfr  and  dx  =  H-dy,  or  dy9  = 

and  dx*  =  —dy2. 
p* 

Substituting,  and  reducing,  we  have, 
dL  = 


xJ°.  Denote  APQB  by  dA  ;  this  is  made  up  of  two  parts, 

the  rectangle,  AR  =  ydx,  and  the  triangle,  RPQ  =  -dydx; 

& 

but  the  latter  is  an  infinitesimal  of  the  second  order,  it  may 
be  therefore  neglected  in  comparison  with  the  former; 
hence,  we  have, 

dA=ydx  .....  (2) 

This  formula  may  be  applied  in  a  manner  similar  to  that 
just  explained.  Thus,  if  it  be  required  to  find  the  differ- 
ential of  the  area  of  a  parabola,  we  have, 


_ 

dA  =  —dy  ;  or,  dA  =  (v%px)dx. 

3°.  Denote  the  surface  generated  by  PQ  by  dS  ;  this 
surface  is  that  of  a  frustum  of  a  cone,  in  which  the  radius 
of  the  upper  base  is  y,  the  radius  of  the  lower  base,  y  +  dyy 
and  the  slant  height,  ydx2  +  dy*.  Hence, 


dS  =      [%«y  +  2<V  +  dy)]  X  Vdxz  +  dy*. 

Neglecting  dy  in  comparison  with  y,  and  reducing,  we 
have, 


dS  =  2«y\/dx*  +  dy* (3) 


APPLICATION    TO' POLAR    CO-ORDINATES. 


93 


To  find  the  differential  of  a  paraboloid,  we  proceed  as 
before.  Substituting  the  values  already  found,  and  re- 
ducing, we  have, 


dS  = 


P2  ',  or,  dS  = 


4°.  Denote  the  volume  generated  by  APQB,  by  dV  ; 
this  volume  is  that  of  a  frustum  of  a  cone,  in  which  the 
radius  of  the  upper  base  is  y,  of  the  lower  base,  y  +  dy^ 
and  the  altitude  dx.  Hence, 


neglecting  dy  in  comparison  with  y,  we  have, 
dV=«y*dx  .....  (4) 

Applying  the  formula  to  the  paraboloid  of  revolution,  we 
have,  as  before, 

•z/3 
3V  =  *  —  dy,  or,  d  V  =  Zepxdx. 

VII.  APPLICATION  TO  POLAR  CO-ORDINATES. 

General  Notions,  and  Definitions. 

52.  In  a  polar  system,  the 
radius  vector  is  usually  taken 
as  the  function,  the  angle  be- 
ing the  independent  variable. 
Let  P  be  any  point  of  the 
curve,  PL,  in  the  plane  of 
the  axes,  OX,  OY  ;  let  0 
be  the  pole,  and  OX  the  ini- 
tial line,  of  a  system  of  polar 
co-ordinates.  Then  will  OP,  Fig.  12. 


YD     X 


94  DIFFERENTIAL   CALCULUS. 

denoted  by  r,  and  XOP,  denoted  by  <p,  be  the  polar  co- 
ordinates of  P,  and  the  equation  of  PL  may  be  written 
under  the  form, 

r  = 


It  will  be  convenient  to  express  the  values  of  9  in  terms 
of  it  as  a  unit  ;  in  this  case,  it  is  laid  off  on  the  directing 
circle,  GS,  in  the  direction  of  the  arrow  when  positive,  and 
in  a-  contrary  direction  when  negative.  Let  GH  be  "the 
measure  of  <p  for  the  point  P,  and  let  HI  be  equal  to  the 
constant  infinitesimal,  dq>  ;  draw  OIQ,  and  with  0  as  a 
centre  describe  the  arc,  PR  ;  then  will  OP  and  OQ  be 
consecutive  radius  vectors,  P  and  Q  will  bo  consecutive 
points,  RQ  will  be  the  differential  of  r,  PQ  will  be  the 
differential  of  the  arc,  POQ  will  be  the  differential  of  the 
area  swept  over  by  the  radius  vector,  and  PQ  prolonged, 
will  be  tangent  to  the  curve  at  P. 

The  line,  BC,  perpendicular  to  the  radius  vector,  OP,  is 
the  movable  axis,  and  the  perpendicular  distance,  OA,  is  the 
polar  distance  of  the  tangent.  Prolong  the  tangent  to  meet 
the  movable  axis  at  B  ;  at  P  draw  a  normal,  and  produce 
it  to  meet  the  movable  axis  at  G  ;  then  is  OB  the  subtan- 
gent,  PB  the  tangent,  00  the  subnormal,  and  PC  tht; 
normal  at  the  point  P. 

Useful  Formulas. 

53.  1°.  Differential  of  the  arc.  Denote  the  differential 
of  the  arc,  PQ,  by  dL  ;  RP  being  infinitesimal  may  be  re- 
garded as  a  straight  line  perpendicular  to  OQ  ;  it  is  equal 
to  rdv  ;  hence, 


dL  =  Vdr*  +  r*d$* (1) 


APPLICATION   TO    POLAR   CO-ORDINATES.  95 

2°.  Differential  of  the  area.  Denote  the  differential  of 
the  area  swept  over  by  the  radius  vector  by  dA  ;  this  is 
made  up  of  the  sector,  OPR,  and  the  triangle,  RPQ  ; 
but,  RPQ  is  an  infinitesimal  of  the  second  order,  and  the 
sector  is  an  infinitesimal  of  the  first  order  ;  neglecting  the 
former  in  comparison  with  the  latter,  and  remembering  that 
the  area  of  a  sector  is  equal  to  half  the  product  of  its  arc 
and  radius,  we  have, 

dA  =  ir'dp  .....  (2) 

3°.  Angle  letween  the  radius  vector  and  tangent.  Denote 
the  required  angle  by  V;  the  angle,  RQP,  differs  from  the 
required  angle,  OPB,  by  an  infinitesimal,  and  may,  there- 
fore, be  taken  for  it;  but  tan  RQP,  equals  RP,  divided  by 
RQ  ;  hence,  we  have, 


=          .....  (3) 
dr 


4°.  Polar  distance  of  the  tangent.  Denote  the  distance 
OA  by  p;  the  triangles,  QPR  and  QOA,  are  similar: 
hence, 

QP  :  RP  :  :  QO  :  OA. 

But,  PO  differs  from  QO  by  an  infinitesimal,  and  may, 
therefore,  be  taken  for  it;  making  this  change,  and  sub- 
stituting for  the  quantities  their  values,  we  have, 


+  r$    :  ry  \\r\p; 
hence, 

= _M* 


9(5 


DIFFERENTIAL   CALCULUS. 


5°.  Formula  /or  subtangenl.  Denote  the  snbtangent  by 
S.T;  in  the  triangle,  POB,  the  perpendicular,  OB,  is 
equal  to  the  base,  OP,  multiplied  by  the  tangent  of  the 
angle  at  the  base;  hence, 


S.T  = 


dr 


(5) 


G°.  Formula  for  subnormal.  Denote  the  subnormal  by 
S.N;  the  triangles,  OPB  and  OOP,  are  similar;  hence, 
tznOCP  =  t'diiOPB;  but,  OC,  equals  OP,  divided  by 
hence, 


7°.  Formula  for  the  radius  of  curvature.  Denote  the 
radius  of  curvature  by  p;  let  P  and  Q  be  two  consecutive 
points  of  the  curve,  KL,  and  let 
PC  and  QC  be  normals  at  these 
points,  meeting  at  C ;  then  will 
C  be  the  centre  of  the  oscillatory 
circle  ;  draw  OM  perpendicular 
to  PC,  it  will  be  parallel  to  the 
tangent  at  P,  and,  consequently, 
the  intercept,  PM,  will  be  equal 
to  the  polar  distance  of  the  tan- 
gent denoted  by  p  ;  draw  0  C, 
OP,  and  0  Q.  From  the  figure,  we  have, 

0  (72  =  rz  f  p2  -  2pP. 

If  we  pass  from  P  to  Q,  r  will  become  r  +  dr,  p  will 
become  p  +  dp,  and  OC  and  p  will  remain  unchanged. 
If,  therefore,  we  differentiate  the  preceding  equation  under 
the  supposition  that  r  and  p  are  variable,  the  resulting 


Fig.  13. 


APPLICATION   TO    EOLAR   CO-ORDINATES.  07 

equation  will  express  a  relation  between  p,  r,  and/*.     Dif- 
ferentiating, \ve  have, 

TflT 

0;     .:  f  =  -      .....  (7) 


8°.  Formula  for  the  chord  of  curvature.  The  chord  ol 
curvature  is  the  chord  of  the  oscillatory  circle  that  passes 
through  the  pole  and  the  point  of  osculation.  Denote  it 
by  C  '  ;  prolong  PC  and  PO  till  they  meet  the  circle  at  R 
and  N,  and  draw  RN.  The  right  angled  triangles,  PNR 
and  PMO,  are  similar,  hence, 

PN  :  PR  :  :  PM  :  PO, 
or, 


nence 


C:  2P  up  :  r; 


=,  or  t'=8p      .....  (8) 
r  dp 


Spirals. 

54.  If  a  straight  line  revolve  uniformly  about  one  of  its 
points  as  a  centre,  and  if,  at  the  same  time,  a  second  point 
travel  along  the  line,  in  accordance  with  any  law,  the  lat- 
ter point  will  describe  a  spiral.  The  part  described  in  one 
revolution  of  the  straight  line  is  a  spire.  The  fixed  point 
is  the  pole.  If  we  denote  the  distance  from  the  pole  to  the 
generating  point  by  r,  and  the  corresponding  angle, 
counted  from  the  initial  line,  by  9,  we  have,  for  the  gen- 
eral equation  of  spirals, 

r  =  /(?) (1) 

When  this  relation  is  algebraic,  the  spiral  is  said  to  be 
algebraic ;  when  this  relation  is  transcendental,  the  spiral 
is  said  to  be  transcendental. 


98  DIFFERENTIAL   CALCULUS. 

Among  algebraic  spirals,  the  most  important  are  the 
spiral  of  Archimedes,  the  parabolic  spiral,  and  the  hyper- 
bolic spiral,  corresponding,  respectively,  to  the  right  line, 
the  parabola,  and  the  hyperbola. 

Spiral  of  Archimedes. 
55.  The  equation  of  this  curve  is, 

r  —  09  .....  (2) 

We  see  that  the  generating  point  is  at  the  pole  when 
the  variable  angle  is  0,  and  that  the  radius  vector  increases 
uniformly  with  the  variable  angle,  being  always  equal  to  o 
times  the  arc  of  the  directing  circle  that  has  been  swept 
over.  Differentiating  equation  (2),  we  have, 

dr  =  ady. 
Substituting,  in  formulas  (1)  to  (8),  Art.  53,  we  find, 


dL  =  acfyl  +  92  ;        S.T=  a^ 
dA  =  4-aV*;  S.N=a; 


~ 


If  we  take  a  straight  line,  whose  equation  is  y  =  ax,  and 
lay  off  the  abscissa  of  any  point  on  the  directing  circle, 
and  the  ordinate  of  that  point  on  the  corresponding 
radius  vector,  the  point  thus  determined  is  a  point  of  th<! 
spiral  of  Archimedes. 


TRANSCENDENTAL  CURVES.  99 

In  like  manner,  we  may  discuss  and  construct  the  para- 
bolic spiral,  whose  equation  is  r2  =  2p$,  and  the  hyperbolic 
spiral,  whose  equation  is  r$  —  m.  The  former  corresponds 
to  the  ordinary  parabola,  and  the  latter,  to  the  ordinary 
hyperbola  referred  to  its  asymptotes. 


VIII.    TRAXSCEXDENTAL  CURVES. 

Definition. 

56.  A  transcendental  curve,  is  a  curve  whose  equation 
can  only  be  expressed  by  the  aid  of  transcendental  quanti- 
ties.    The  cycloid,  and  the  logarithmic  curve  are  examples 
of  this  class  of  lines. 

The   Cycloid. 

57.  The  cycloid  is  a  curve  that  may  be  generated  by  a 
point  in  the  circumference  of  a  circle,  when   that  circle 
rolls  along  a  straight  line.     The  point  is  called  the  genera- 
trix, the  circle   is   called  the  generating   circle,  and   the 
straight  line  is  the  base  of  the  cycloid.     The  curve  has  an 
infinite   number  of  branches,  each   corresponding  to  one 
revolution  of  the  generating  circle. 

To  find  the  equation  of  one  branch,  A PM ;  let  A  be  tne 
origin,  0  the  centre  of  the  generating  circle,  and  P  the 
place  of  the  generatrix  after  the  circle  has  rolled  through 
the  angle  KCP,  denot- 
ed by  <p ;  denote  the  co- 
ordinates, AL  and  LP, 
by  x  and  ?/,  and  let  the 
radius  of  the  generating  AL 
ci  rcle  be  represen  ted  by  r.  Fig.  14. 

From  the  figure,  we  have, 

=  AK-  LK. 


100  DIFFERENTIAL   CALCULUS. 


Bat,  AK  is  equal  to  the  arc  /fP,  and  KP  is  the  arc 
whose  versed  sine  is  y,  to  the  radius  r,  or  r  times  the  arc 

7/ 

whose  versed  sine  is  —  ,  to  the  radius  1  ;  LK  is  equal  to 

P/>,  which,  from  a  property  of  the  circle,  is  equal  to 
=t  V(2r  —  y)y,  the  upper  sign  corresponding  to  the  case 
in  which  P  is  to  the  left  of  KE,  and  the  lower  sign  to  the 
case  in  which  P  is  to  the  right  of  KE  ;  and  AL  is  equal 
to  x.  Substituting  these  values,  and  reducing,  we  have, 


j       nt 

x  =  rversin"    - 


which  is  the  equation  sought.  From  it  we  see  that  y  can 
never  be  less  than  0,  nor  greater  than  2r;  we  see,  also,  that 
y  is  equal  to  0,  for  x  =  0,  and  for  x  =  2<7rr,  and  that  for 
every  value  of  y,  there  are  two  values  of  x,  one  exceeding 
•xr  as  much  as  the  other  falls  short  of  it. 
Differentiating  and  reducing,  we  have, 

dx  - 

(Mi    -    — 


which  is  the  differential  equation  of  the  cycloid.     From  it 
we  deduce  the  equation, 

~r  —  ^  A/  --  1>  whence,  -~  =  --  -. 
dx  y     y  dxz  y2 

From,  the  first,  we  see  that  the  tangent  to  the  curve  is 
vertical  for  x  —  0,  and  x  =  2tfr,  and  that  it  is  horizontal 
for  x  =  *r  ;  from  the  second,  we  see  that  the  curve  is  always 
concave  downward. 

Substituting,  in  formulas  (6)  and  (7),  Art.  32,  we  have. 


=±--===,  and 


TRANSCENDENTAL   CURVES. 


101 


The  Logarithmic  Curve. 

58.  The  logarithmic  curve,  is  a  curve  in  which  any 
ordinate  is  equal  to  the  logarithm  of  the  corresponding 
abscissa.  Its  equation  is,  therefore, 

y  —  logs,  or  y  =  M.lx; 

in  which  M  is  the  modulus  of  the 
system. 

If  M  >  0,  y  will  be  negative  when 
x  <  1,  positive  when  x  >  1,  0  when 
x  =  1,  and  infinite  when  x  —  0. 

If  M  <  0,  y  will  be  positive  when 
x  <  1,  negative  when  x  >  1,  0  when  x  =  1,  and  infinite 
when  x  =  0. 

In  no  case  can  a  negative  value  of  -a?  correspond  to  a 
point  of  the  curve. 

Differentiating,  we  find, 


Fig.  15. 


~i~ 

dx 


-7   „ 

dx2 


The  first  expression  has  the  same  sign  as  My  and  varies 
inversely  as  x.  Hence,  when  M>  0,  the  slope  is  positive, 
when  M  <  0,  the  slope  is  negative,  and  in  both  cases  tin- 
curve  continually  approaches  parallelism  with  the  axis 
of  x. 

The  second  expression  has  a  sign  contrary  to  that  of  M, 
and  varies  inversely  as  the  square  of  x.  Hence,  when  M  >  0, 
the  curve  is  concave  downward,  as  KDC,  and  when  M  <  0, 
it  is  concave  upward,  as  KDW. 


PART  III. 

INTEGEAL  CALCULUS. 


Object  of  the  Integral  Calculus. 

59.  The  object  of  the  integral  calculus  is  to  pass  from  a 
given  differential  to  a  function  from  which  it  may  have 
been  derived.  This  function  is  called  the  integral  of  the 
differential,  and  the  operation  of  finding  it  is  called  inte- 
gration. The  operation  of  integration  is  indicated  by  this 

sign, /,  called  the  integral  sign.    Integration  and  differ 

entiation   are   inverse  operations,   and  their   signs,  when 

placed  before  a  quantity,  neutralize  each   other;   thus, 
,  % 

dx  =  x. 

A  constant  quantity  connected  with  a  function  by  the  sign 
of  addition,  or  subtraction,  disappears  by  differentiation, 
hence,  we  must  add  a  constant  to  the  integral  obtained; 
and  inasmuch  as  this  constant  may  have  any  value,  it  is 
said  to  be  arbitrary.  The  integral,  before  the  addition  of 
the  constant,  is  called  incomplete,  after  its  addition  it  is 
said  to  be  complete.  By  means  of  the  arbitrary  constant, 
as  we  shall  see  hereafter,  the  integral  may  be  made  to 
satisfy  any  one  reasonable  condition. 

Nature  of  an  Integral. 

60.  The  differential  of  a  function  is  the  difference 
Ix'tweon  two  consecutive  states  of  that  function;  hence, 


NATURE   OF   AN   INTEGRAL.  103 

any  state  of  the  function  whatever,  is  equal  to  some  pre- 
vious, or  initial  state,  plus  the  algebraic  sum  of  all  the 
intermediate  values  of  the  differential.  But  this  state  is 
the  integral,  by  definition  ;  hence,  if  the  arbitrary  constant 
represents  the  initial  state,  the  incomplete  integral  repre- 
sents the  algebraic  sum  of  all  the  differentials  from  the 
initial  state  up  to  any  state  whatever,  and  the  complete 
integral  represents  the  initial  state,  plus  the  algebraic  sum 
of  all  the  differentials  from  that  state  up  to  any  state 
whatever. 

Before  any  value  has  been  assigned  to  the  constant,  the 
integral  is  said  to  be  indefinite ;  when  the  value  of  the 
constant  has  been  determined,  so  as  to  satisfy  a  particular 
hypothesis,  the  integral  is  said  to  ^particular  ;  and  when 
a  definite  value  has  been  given  to  the  variable,  this  integral 
is  said  to  be  definite. 

The  values  of  the  variable  that  correspond  to  the  initial 
and  terminal  states  of  a  definite  integral  are  called  limits, 
the  former  being  the  inferior,  and  the  latter  the  superior 
limit ;  the  integral  is  said  to  lie  between  these  limits.  The 
value  of  tlip  definite  integral  may  be  found  from  the  indefi- 
nite integral  by  substituting  for  the  variable  the  inferior 
and  superior  limits  separately,  and  then  taking  the  first 
result  from  the  second.  The  first  result  is  the  integral 
from  any  state  up  to  the  first  limit,  and  the  second  is  the 
integral  from  the  same  state  up  to  the  last  limit;  hence, 
the  difference  is  the  integral  between  the  limits.  The 

Cb  '       T,'  i, 
symbol  for  integrating  between  the  limits  is  I    >  m  wnicn  a 

is  the  inferior,  and  I  the  superior  limit. 

This  article  will  be  better  understood  when  we  come  to 
its  practical  application  in  Articles  78  and  following. 


104  LNTEGKAL   CALCULUS. 

Methods  of  Integration  5  Simplifications. 

61.  The  methods  of  integration  are  not  founded  on  pro- 
cesses of  direct  reasoning,  but  are  mostly  dependent  on 
formulas.     The  more  elementary  formulas  are  deduced  by 
reversing  corresponding  formulas  in  the  differential  ,cal- 
culus;  the  more  complex  ones  are  deduced  from  these  by 
various  transformations  and  devices,  whereby  the  expres- 
sions to  be   integrated  are  brought   under  some  known 
integrable  form. 

It  has  been  shown  (Art.  9),  that  a  constant  factor  re- 
mains unchanged  by  differentiation;  hence,  a  constant 
factor  may  be  placed  without  the  sign  of  integration.  It 
was  also  shown  in  the  same  article,  that  the  differential  of 
the  sum  of  any  number  of  functions  is  equal  to  the  sum 
of  their  differentials;  hence,  the  integral  of  the  sum  of 
any  number  of  differentials  is  equal  to  the  sum  of  theii 
integrals.  These  principles  are  of  continued  use  in  inte- 
gration. 

Fundamental  Formulas. 

62.  If  we  differentiate  the  expression -,  we  have, 


reversing  the  formula,  and  applying  the  integral  sign  to 
both  members, 


Remembering  that  the  signs  of  integration  and  lifferentia* 


METHODS   OF  INTEGRATION.  105 

tion  neutralize  each  other  (Art.  50),  and  adding  a  constant 
to  complete  the  integral,  we  have, 


[1] 


Hence,  to  integrate  a  monomial  differential,  —  drop  the 
differential  of  the  variable,  add  1  to  the  exponent  of  the 
variable,  divide  the  result  ~by  the  new  exponent,  and  add  a 
constant. 

This  rule  fails  when  n  =  —  1  ;  for,  if  we  apply  it,  we  get 
a  result  equal  to  oo  ;  in  this  case  the  integration  may  be 
effected  as  follows  : 

From  Art.  15,  we  have, 


7/  7  \  _   , 

d(alx)  =  a  —  =  ax     dx  ; 
x 


reversing  and  proceeding  as  before,  we  have, 

fax~ldx  =  of—  =  alx+C  .....  [2] 
From  Art.  16,  we  have, 


reversing  and  proceeding  as  before,  we  have, 


L3J 


In  like  manner,  reversing  the  formulas  in  Article  17,  let- 
tered from  a  to  lit  we  have, 

I  cosxdx  —  smx  +  C  ............  .  [4] 


5* 


106  INTEGRAL   CALCULUS. 

I  —  Biaxdx  —  cosz  +  O  ..........  [5] 

r_dx  =         0  ............  [6j 

J  coszx 

-^L  =  cot*  +  (7  ...........  [7] 

sm2a; 

I  siuxdx  =  versing  +  0  ..........  [8] 

/  —  cosa;^  =  coversinz  +  C  .......  [9] 

Jt&nxsQcxdx  =  seai?  +  C  .........  [10] 

/  —  cota;coseca;^  =  cosecx  +  0  .....  [11  J 


In  like  manner,  from  the  formulas  in  Art.  18,  lettered  from 
a'  to  h',  we  deduce  the  following: 

r-M=  =  S^-ly  +  G  ........  [12] 

JVl-y* 

-^=  =  coS-V+<7  .......  [13] 


[15] 


METHODS  OF  INTEGRATION.  107 

/    /===  =  versiu~V  +  C  ......  [161 

J     ^y    _    y2 

/-     .  ___  —  =  coversin~V  +  O  .  .  .  [17] 
V2    -    2 


(7 


--  -p-  -  =  cosec-V  +  C  ----  [19J 

V*  -  1 


If  in  formulas  (12),  (13),  (14),  (15),  (18),  (19),  we  make 


Ix      ,  ,      , 

»/  =  —  ,  whence  dy  =  —  ,  and  reduce,  we  hare, 


d 

v* 


dx  1   .   _   bx 


/cfo  1        _  ifz 

^-^  =  S5ten    ¥+c 

/•__*=  loot'1  52+  (7.     .[23] 
/       a2  4-  ^2a2       aJ  a 

=  lsec-l^+a.,     .[241 


—  a*      a 


108  INTEGKAL  CALCULUS. 


From  (16)  and  (17),  by  substituting  —  j-  x  for  y,  and  re- 
ducing, we  have, 


C 
J 


dx 


=  1  versing*  +  C  .....  [26J 
* 


1  .   _i238 

=  T  coversm    x  —  r-  x  +  (7  .  .  [271 
2 


Formulas  (1)  to  (19)  are  the  elementary  forms,  to  one 
of  which  we  endeavor  to  reduce  every  case  of  integration  ; 
the  processes  of  the  integral  calculus  are  little  else  than  a 
succession  of  transformations  and  devices,  by  which  this 
reduction  is  affected. 

In  the  following  examples,  and,  as  a  general  thing, 
throughout  the  book,  the  incomplete  integral  is  given,  it 
being  understood  that  an  arbitrary  constant  is  to  be  added 
in  every  case. 

EXAMPLES. 
Formulas  (1)  to  (3). 

1.  dy  =  tx*  dx.  Ans.  y  =  -j-, 

i  24 

2.  dy  =  :z*dx.  Ans.  y  =  -lx*. 

3.  dy  =  3x~*cfa.  An*,  y  =  —  x      , 


METHODS  OF  INTEGRATION.  10!) 

4.  dy  =  Zx^dx.  Ans.  y  =  ~x  *. 

2    -4  10  -i 

5.  dy  =  -    x     dx.  Ans.  y  =  -x     . 


e.     =  -«£  —  -«  fo 


8.  dy  = 

A 


9.  r??y  =  3x~<lx.  Ans.  y  -  Six. 

f*        K 


.cosa? 


11.  dy  =  eBinxcosxdx.  Ans.  y  = 

12.  dy——  ecosxsinxdx.  Ans.  y  = 

Formulas  (4)  to  (11). 
14.  dy  =  -  lsin(o;2)  X  xdx.  Ans.  y  =  Tcos(z2). 

&  • 

^  r\r\c*  2/1  nr* \ 


.  y  =  lcot(3.r»). 


110 
17.  dy  =  &iu(ax)dx. 


INTEGRAL   CALCULUS. 


Ans.  y  =  —  versing). 


18.  dy  —  —  cosf  —  x2  }  X  xdx.  Ans.  y—  coversinf  —  xz  Y 

Formulas  (12)  to  (19). 

19.  dy  =  — — — .  Ans.  y  =  sin"1(2^). 


20.  dy  = 


Ans.  y  =  sin    i(xt) 


a&dx 


IF  vT 


Ans.  y  =  ^sin     (x 

o 


22.  J    =  — 


cfa; 


xdx 


Zdx 


y  =  cos"1 


, 

=    tan 


=  2cot 


25.  <     = 


y  2  — 


METHODS  OF   INTEGRATION. 

Formulas  (20)  to  (25). 
;    (a  =  v2j  "b  =  3). 


1H 


27.  rf    = 


;  (a  = 


Ans.  y  =  -tan    x  ~. 


=  cot" 


=  -— rrCOSCC 


112  INTEGRAL  CALCULUS. 

32.  dy  —  - — ?—.  Ans.  y  = 


33.  dy  =          .          Ans.  y  =  — -cosec" 

"  /T^       n  9  .    /«-» 


V3 

Formulas  (26)  and  (27). 
34.  dy  =  — = ;    (a  =  2,  J  =  3). 


=  —  versm 

o 


—  Zaxdx        .  , 

85.  cfr=    _  —  —  -;  («=i,  a= 


EXAMPLES  IN  SUCCESSIVE   INTEGRATION. 

36. 


Dividing  by  dx2, 

VT  =  axdx,  or  d(  -~]  =  axdx. 
dx2  \dxzj 

Integrating,  we  have, 

d*y  _  ax* 

d&    ~T 

Multiplying  by  dx,  and  integrating  again,  we  find, 
dy      ax3 

j.  =  _  +  Cx+(7. 


METHODS   OF   INTEGRATION.  113 

Multiplying  again  by  dx,  and  integrating, 


37.  d*>y  =  —  x~*d 
As  before,  we  have, 
»-3 


<&»-*-  —  +  ux  + 


and  finally, 


1,         Cx*       C'x2 
=  -fa  +  -g-  +  -y- 


38.  6?3y  =  mdx*. 

mx* 


„ 

Ox 
o  /« 


39.  t?3y  =  x* dx3. 


105 
40.  J2?/  =  sx*dx*. 


114  INTEGRAL   CALCULUS. 

Integration  by  Parts. 

63.  If  u  and  v  are  functions  of  x,  we  have,  from  Art.  10, 

d(uv)  =  udv  +  vdu. 
Integrating  and  transposing,  we  have, 

I  udv  =  uv  —  t  vdu [28] 

This  is  called  the  formula  for  integration  by  parts;  it 
enables  us  to  integrate  an  expression  of  the  form  udv 
whenever  we  can  integrate  an  expression  of  the  form  vdu. 
It  is  much  used  in  reducing  integrals  to  known  forms. 

EXAMPLES. 

1.  dy  =  xlx  dx  / 
Let  Ix  =  Uj  and  xdx  =  dv  / 

,        dx        ,          a* 
/.  du  =  — ,  and  v  =  — -. 

X  A 

Substituting  in  the  formula,  we  have, 

__  x2 lx  _  rxdx  _  x*lx  __  x2 
~2       J  ~2~       ~2        T' 


—  ^2  =  u,  and  —-  =  dv; 


,  xdx  1 

.".  du  —  --  -  ---  ,  and  v  =  —  • 


METHODS   OP   INTEGRATION.  115 

hence,  we  have,  from  the  formula, 


Vl  -  xz        r      dx                 Vl-x*        .       « 
y  = /  — = sm"1^. 

X  J    A/i    _  3.8  iC 


Additional  Formulas. 

64.  1°.  Formula  (1)  may  be  extended  to  cover  the  case 
of  a  binomial  differential  of  the  form, 


in  which  the  exponent  of  the  variable  without  the  paren- 
thesis is  1  less  than  the  exponent  of  the  variable  within. 

Let  a  +  bxn  =  z;    .-.  bnxn~ldx  =  dz,  or  xn~ldx  =  ^. 

bn 

Substituting,  and  integrating,  we  have, 


y  = 
Keplacing  z  by  its  equal,  (a  +  &sn),  we  have, 

J  '(.  +  fc^^i*  =  1  +  a 


Hence,  to  integrate  a  binomial  differential  of  the  proposed 
form,  add  1  to  the  exponent  of  the  parenthesis,  divide  the 
result  by  the  new  exponent  into  the  product  of  the  coefficient 
and  the  exponent  of  the  variable  within  the  parenthesis. 

It  is  to  be  observed  that  a  constant  factor  may  be  set 
aside  during  the  process  of  integration,  and  then  intro- 
duced, as  explained  in  Article  Gl. 


116 


CALCULUS. 


EXAMPLES. 


* 


1.  dy  =  (1  +  x*)*xdx. 

xdx 

2.  dv  — 


4.  dy  —  (2 

5.  dy  =  - 


Ans.  y  = 


Ans.  y  = 


Ans.  y  =  —  (7 

lo 


.  ;y  =       (2 


2°.  The  preceding  formula  fails  when  p  =  —  1  ;  in  that 
case,  we  have, 


. 

bnJ   z       In 


Replacing  z  by  its  value, 


EXAMPLES. 


[30  1 


1.  dy  =  3(3  -f 

*fc 

2.  r/?x  = 


Ans.      •-= 


re  +  a 


0     7 
3.  r     = 


. 

x  +  4 


.  y  =  l(x  +  a), 
Ans.  y  =  2  7a  +  41 


METHODS   OF   INTEGRATION.  11? 

When  the  differential  expression  is  a  fraction  in  which 
the  numerator  is  equal  to  the  differential  of  the  denomi- 
nator multiplied  by  a  constant,  its  integral  is  equal  to  that 
constant  into  the  Napierian  logarithm  of  the  denominator. 

4.  dy  =  ^Ar  Ans'  V  =  "  3  l(a*  ~  x^' 


3x2 


Ans.  y  =  l(x*  +  x2  +  x  -f  1) 
3°.  Let  it  be  required  to  integrate  the  expression. 

dx 

\/x2  dc.  a2 
assume, 

xz  =t  az  =  z2 ,  whence,  xdx  =  zdz  ; 

adding  zdx  to  both  members  and  factoring,  we  have, 

(x  +  z} dx  =  z(dx  -f  dz)  ; 
hence, 

dx  _         dx         _  dx  4-  dz  _  d(x  -\-  z)  > 

applying  the  principle  just  deduced,  we  have, 


Replacing  z  by  its  value,  we  have,  finally, 

/* — ^ =  i(x  +  V^io5)  +  C [31] 

J   \/x2  ±  a2 


118 


INTEGRAL   CALCULUS. 


EXAMPLES. 


1     //     -  - 

—  7  ~ 


-  5 

4°.  Let  it  be  required  to  integrate  the  expression, 

dx 


since,  dx  =  d(a  +  #),    and    2aa;  +  a;2  =  (a  -f  ^)2  —  a*, 
we  have, 

/(fa         _  /»       d(a  +  x) 
V^ctx  +  x*J  >\/a  +~zz  —  a? 


which  can  be  integrated  by  Formula  (31);  hence, 


/———^=  =l\a 
V'2ax  +  x2         ( 


...  [32J 


EXAMPLES. 

dx 


X 


Ans.  y  =  l(x  +  -  +  -v/|F+75 


METHODS   OF   INTEGRATION-.  119 


Vbx  +  9z2 
5°.  Let  it  be  required  to  integrate  the  expression, 

dx 

dy  =  —  --  -: 
az  —  x9 

Factoring,  we  have, 

dx  1  j    dx  dx    ) 

a2  —  x*  ~2a(  a  +  x  +  a  —  x  \  ' 

Integrating,  we  have, 


But  the  difference  of  the  logarithms  of  two  quantities  is 
equal  to  the  logarithm  of  their  quotient;  hence, 

/V^-i  =  lj«±£+0.          .[33| 
J  a*  —  xz       2a    a  —  x 

and  in  like  manner, 

.....  [34| 


a    x  +  a 

EXAMPLES. 

dx  1  ,  3  -f  x 


dx  1    x  —  2 


120  INTEGRAL   CALCULUS. 

dzy 
S.  Given  -j-^  =  y,  to  find  the  relation  between  y  and  x. 

Multiplying  both  members  by 


Integrating, 

dt/2  dii 

'  ' 


T?  -  y        » 

dx* 
Integrating  by  Formula  31, 

x  = 

dzs 
4.  Given  -j-^  =  —  nzs,  to  find  tlie  relation  between  a 

and  t,  t  being  the  independent  variable. 
Multiplying  by  2ds, 


•dP 

In  tegrating, 

^s_  22  ds 

dt^~  ~  V0 -»»*»' 

Integrating  by  Formula  (20), 

,      1    .   _i  ns        ~. 

t  =  -  sin    L  —^  +  C'. 
n 


Rational  and  Entire  Differentials. 

65.  An  algebraic  function  is  rational  and  entire  when  it 
contains  no  radical  or  fractional  part  that  involves  the 
variable.  If  the  indicated  operations  be  performed,  every 


METHODS   OF   INTEGRATION.  121 

rational  and  entire  differential  will  be  reduced  to  the  form 
of  a  monomial,  or  polynomial  differential,  each  term  of 
which  can  be  integrated  by  formulas  (1)  or  (2).  Thus, 
if  the  indicated  operations  be  performed,  in  the  expression, 


we  have, 

dy  =  If  8x«  -  ±x*  +  6z*  -  3z3  \dx. 

Hence,  by  integration, 


2  6    .        3     . 

=          *  X    +     X     ~X 


In  like  manner,  any  rational  and  entire  differential  may 


be  integrated. 


Rational  Fractions. 


66.  A  rational  fraction,  is  a  fraction  whose  terms  are 
rational  and  entire.  When  a  rational  fraction  is  the  differ- 
ential of  a  function,  and  its  numerator  is  of  a  higher  degree 
than  its  denominator,  it  may,  by  the  process  of  division,  be 
separated  into  two  differentials,  one  of  which  is  rational 
and  entire  and  the  other  a  rational  fraction,  in  which  the 
numerator  is  of  a  lower  degree  than  the  denominator.  The 
former  maybe  integrated  as  in  the  last  article;  it  remains 
to  be  shown  that  the  latter  can  be  integrated  whenever 
the  denominator  can  be  resolved  into  binomial  factors  of 

the  first  degree  with  respect  to  the  variable.     The  method 

G 


122  INTEGRAL   CALCULUS. 

of  integration  consists  in  resolving  the  differential  coeffi- 
cients into  partial  fractions,  by  some  of  the  known  methods, 
then  multiplying  each  by  the  differential  of  the  variable 
and  integrating  the  polynomial  result.  There  are  four 
cases,  depending  on  the  method  of  resolving  the  differen- 
tial coefficient  into  partial  fractions.  Each  case  will  be 
illustrated  by  examples. 

First  Case. —  When  the  binomial  factors  of  the  denomi- 
nator are  unequal. 

Let  us  have, 

..  \ dCC  *~~"  0  )  CCCC  &Q5  ~~~  0 

ay  = — = r-j ; — -, dx. 

Assume  the  identical  equation, 

2x-5  =     A       t      B      .       V 


(x  -  i)  (x  _  2)  (x  -  3)  ~~  x  -  1      x  -  2      x-3  ' 

Clearing  of  fractions. 
2a?  —  5  =  A(x  -2)  (x  —  3)  +  B(x  —  1)  (x  -  3) 

+  C(x  -  1)  (x  -  2) (2) 

In  order  to  find  A,  B,  and  0,  we  might  perform  the 
operations  indicated  in  (2),  equate  the  coefficients  of  the 
like  powers  of  x,  and  solve  the  resulting  equations;  but 
there  is  a  simpler  method  in  cases  like  this,  depending  on 
the  fact  that  (2)  is  true  for  all  values  of  x. 

0 

Making  x  =  1,  we  find,  —  3  =     2.4; 

Making  x  =  2,  we  find,  —  1  =  —  B;    .-.  B  =       1. 

Making  x  =  3,  we  find,        1  =      2(7;     /.  C  —       «• 


METHODS   OF   INTEGRATION. 


Substituting  these  in   (1),  multiplying   by  dx,  and  in 

tegrating  by  Formula  30,  we  have, 

• 

(2x  -  5)<fa  _3    f  dx         f  dx 

jc  -l)(x-  2)  (x  -  3)  ~~       %Jz>-l    Jx-% 


Reducing  the  result  to  its  simplest  form,  in  accordance 
with  the  elementary  principles  of  logarithms,  we  have, 
finally, 


(x  -  1) 

EXAMPLES. 

5x  +  l)dx  _      (5x  +  l)dx 


x  -  2  "  (x  -  1)  (a?  +  2)* 


a;  +  2)  =  ?(a?  -  I)    X  (x 


(a;  -  l)dx 
y  ~  x*  +  6x  +  8  "  (x  +  2)  (x  +  4)' 


y  = 


x  +  2)* 


"      x3+x2—2x      x(x  —  1)  (x  +  2)* 


4.  dv  = 


x(x-l)(x  +  1)' 

<,   y  —  l(x*   —  x). 


124  INTEGRAL   CALCULUS. 

Second  Case. —  When  some  of  the  binomial  factors  are 
equal. 

In  this  case  "there  are  as  many  partial  fractions  as 
there  are  binomial  factors  in  the  denominator;  but  the 
denominators  of  those  corresponding  to  factors  of  the 

form   (x  —  a)my   are   respectively   of  the   form   (x  —  a)m, 
(x  -  a)  m~1)  (x  -  a)m~2,  etc.,  down  to  x  -  a. 
Let  us  assume, 

*,  -          (x*  +  *)<** 


assuming  the  identical  equation, 

(x  -  2p  (x  -  1)  =  (x^~2^  +  x-2  +  ~x~^\ W 

and  clearing  of  fractions,  we  have, 

z2  +  x  =  A(x  —  1)  +  B(x  —  2)  (x  —  1)  4-  C(x  —  2)2...  (2) 

Here  again  we  might  find  the  values  of  A,  B,  a  id  0,  by 
the  ordinary  method  of  indeterminate  coefficients ;  but  it 
will  be  simpler  to  proceed  as  follows : 

Making  x  =  2,  we  find,  A  =  6 ; 
Making  x  =  1,  we  find,  (7=2; 

giving  to  A  and  C  their  values  in  (2),  and  then  making 
y>  =  0,  we  find,  0  =  -  6  4-  2£  +  8     .-.  B  =  -  1. 
Substituting  in  (1)  and  multiplying  by  dx,  we  have, 


(x*  4-  x)dx  6dx  dx          2dx 

_2)~2~(^—  1)  ~  (x-  2)2  ~  x  —  2      x  — 


METHODS   OF   INTEGRATION.  125 

The  first  partial  fraction  can  be  integrated  by  Formula 
29,  and  the  others  by  Formula  30  ;  hence, 


x  —  2  x  —  Z         x  —  2 


EXAMPLES. 


.  —  --  —  -  . 

X3  (X  —  1) 

In  this  example  the  equal  binomial  factors  are  of  the 
form  (x  —  0)  or  x,  and  the  assumed  identical  equation  is, 

x*  -2      _  A       B_       C         D 
x*(x-l)~  x*  +  x2  +  z  +  x^l  ' 

Clearing  of  fractions,  and  performing  indicated  operations, 
we  have, 


xz  -  2  =  Ax  —  A  +  ^2  -  Bx  +  Cx3  —  Cx*  +  Dx*. 

Equating  the  coefficients  of  like  powers  in  the  two  mem- 
bers, and  solving  the  resulting  group,  we  have,  A  =  ^, 
J3  =  2,  0  =  1,  and  D  —  —  1  ;  substituting  in  (1),  and 
proceeding  as  before,  we  have, 


x 


j     _  _  xdx 

= 


126  INTEGRAL  CALCULUS. 

Third  Case.  —  Wlien  some  of  the  factors  are  imaginary, 
~but  unequal. 

In  this  case  the  product  of  each  pair  of  imaginary 
factors  is  of  the  form  (x  —  a)2  +  bz  ;  instead  of  assuming 
a  partial  fraction  for  each  imaginary  factor,  we  assume 
for  each  pair  of  such  factors  a  fraction  of  the  form, 

A  +  Bx 


xa*x 


Assume  the  identical  equation, 

x  A  -f  Bx          O 


(1) 


(a.  +  i)  (aj»  +  l)        xz  +  l        a:  +  1  ' 
Clearing  of  fractions,  and  reducing, 

x  =  Bx*  +  (^  +  5)a;  +  A  +  Cx*  +  C (2) 

Equating  the  coefficients  of  the  like  powers  of  x  in  the  two 
members,  and  solving  the  resulting  equations,  we  have, 

Substituting  in  (1),  and  multiplying  by  dx,  we  have, 
xdx  _  1  (1  +  x)dx  __  1     dx 


1  /     f/a;  .T<:fe  ^/./:    \ 

2  U*  +  1  +  ^Tl  ""  a;  +  17 


Integrating  by  Formulas  (14)  and  (30),  and  reducing,  we 
have, 


METHODS   OF   INTEGRATION".  12? 

Fourth  Case.  —  When  some  of  the  binomial  factors  are 
imaginary  and  equal. 

In  this  case,  the  denominator  will  contain  one  or  more 

factors  of  the  form  [(a;  —  a)9  +  &2]n;  in  determining  th<» 
partial  fractions,  we  combine  the  methods  used  in  the 
second  and  third  cases. 

Let  us  assume, 

x3dx 


Assume  the  identical  equation, 

r3  Ax  +  B  Cx  +  D 


Clearing  of  fractions,  and  proceeding  as  before,  we  find, 
A  =  2,  B  =  -  4,  C  =  1,  and  D  =  2. 

Substituting  these  in  (1),  multiplying  by  dx,  and  separating 
the  fractions,  we  have, 


.    _  2xdx  kdx  xdx 

.J  /-v.2  O 


2)2       (x2  -  2.-C  +  2)2       x*  -  2x  +  2 

(2) 

' 


Making  (x  —  I)2  +  1  =  z2  +  1,  whence  x  —  1  =  z,  and 
dx  =  dz,  we  have, 

,   _  Z(z  +  l)dz  ±dz          (z  +  1)dz 

or, 


2^  zdz  Sdz 

y  ~  (^T"i)8     J^Ti8    *»  + 1    2*  +  r 


INTEGRAL   CALCULUS. 

,  third,  and  fourth  terms  can  be  integrated  by 
known  formulas ;   the  second  is  a  particular  case  of  the 

form ,  which  can  be  integrated  by  the  aid  of 

(*•  +  D» 

Formula  D,  yet  to  be  deduced. 

Integrating  i\\Q  first,  third,  and  fourth,  and  indicating 
the  integration  of  the  second,  we  have, 


V  =  -  W~T  -  a/TTl      +  \ 


From  what  precedes  we  infer  that  all  rational  differen- 
tials are  integrable;  consequently,  all  differentials  that 
can  be  made  rational  in  terms  of  a  new  variable  are  also 
integrable. 


Integration  by  Substitution,  and  Rationalization. 

67.  An  irrational  differential  may  sometimes  be  made 
rational,  by  substituting  for  the  variable  some  function  of 
an  auxiliary  variable ;  when  this  can  be  done,  the  integra- 
tion may  be  effected  by  the  methods  of  Articles  65  and  66. 
When  the  differential,  can  not  be  rationalized  in  terms  of 
an  auxiliary  variable,  it  may  sometimes  be  reduced  to  one 
of  the  elementary  forms,  and  then  integrated.  The  method 
of  proceeding  is  best  illustrated  by  examples. 


When  the  only  Irrational  Parts  are  Monomial. 

68.  When  the  only  irrational  parts  are  monomial,  a  dif- 
ferential can  be  made  rational  by  substituting  for  the 
primitive  variable  a  new  variable  raised  to  a  power  whose 


METHODS   OF   INTEGRATION.  129 

exponent  is  the  least  common  multiple  of  all  the  indices 
in  the  expression. 

(**  —  x*)dx  ,  x 

Let  dy  =  *  --  —  .....  (1) 


The  least  common  multiple  of  the  indices  is  6  ;  making 

x  =  z*,  we  have,  x*  =  z3,  x*  =  24,  a?*  =  z2,  and  cfo  = 
these  substituted  in  (I),  give, 


*9   _  * 


Performing  the  division  indicated,  we  have, 

dy  =  —  6  (zi  —  z6  —  z5  +  z4  +  2s  —  z*  — 


Integrating  by  known  methods,  and  replacing  z  by  its 
value,  a;*,  we  have,  finally, 

3464  6484 

y  =  —  -gfi  +  y  «<>  +  a?  -  ^  ««  —  -  %* 


—  6      -  31(1  +  o) 

When  the  only  irrational  parts  are  fractional  powers 
of  a  binomial  of  the  first  degree,  the  differential  can  be 
rationalized  by  the  same  rule,  and  consequently  can  be 
integrated. 

Let  us  take  the  expression, 


dy  -=  (x  +  Vx  +  2  +  ^aT+I)  dx 


130  INTEGRAL   CALCULUS. 

Assume, 

x  +  2  =  z9,  whence,  x  =  z*  —  2,  and  dx  —  v«-u,*. 

Substituting  in  (2),  we  have, 


Integrating,  and  substituting  for  z  its  value,  (a;  -f  2)  %  we 
have, 

y  =  I  (x  +  2)*  -  *(x  +  2)  +  |  (x  +  2)*+  J  (z  +  2)*. 


Binomial  Differentials. 

69.  Every  binomial  differential  can  be  reduced  to  the 

Form, 

r 

rfy  =  -4  (a  +  fa*)*xm~ldx  .....  (1) 

in  which  m,  n,  r,  and  5  are  whole  numbers,  and  n  positive, 

Thus,  the  expression,  (x~  *  -f  8flT*)~*aAfe,  can  be  re- 

i_2     1 
uu^u  ^  uu*  xv^*^  VJ.  ,  /v^  )   %"^cfe  by  simply  removing 

the  factor  x~~k  from  under  the  radical  sign.  If,  in  this 
result,  we  make  x  =  z*,  whence,  dx  =  ±z3dz,  it  will  be- 

come, 4(1  +  2z)~*z5dz,  which  is  of  the  required  form.  In 
like  manner,  any  similar  expression  may  be  reduced  to 
that  form.  The  constant  factor,  A,  may  be  omitted  during 

integration,  and  the  exponent  of  the  parenthesis,  -,  may 

s 

be  represented  by  a  single  letter  p,  as  is  done  in  Article  70. 
After  integration,  the  factor  may  be  replaced,  and  the  value 

of  -  may  be  substituted  for  p. 


METHODS   OF   INTEGRATION.  131 


FIRST   CRITERION. 

Form  (1)  may  be  integrated  by  some  of  the  methods 

T 

explained  in  Articles  Go  and  66,  when  -  is  a  whole  number. 


SECOND   CRITERION. 

Form  (1)  can  be  integrated  when  —  is  a  whole  number. 

1 

For  let  us  assume  (a  4-  bxn)  =  z*9  or,  z  =  (a  +  lxn)*;   we 
have,  by  solution  and  differentiation, 

™ 


=\-r) ;a 

and, 

m 

m—l  ,         *   /«*  —  a\~n        s—l-r 

xm    Ldx  =  ^—  (  — 7 —  )         z      dz. 
bn\     1}     ) 

Substituting  in  (1),  we  find, 

\  /  ~  bn\      b     )  •  •  •  V.  / 


Which  expression  is  integrable,  as  was  shown  above,  when 


—  is  a  whole  number. 
n 


THIRD  CRITERION. 

in      T 
Form  (1)  can  be  integrated  when  — I-  -,  is  a  whole 

number. 

For  let  us  assume, 


132  IKTEGKAL   CALCULUS. 

Solving  and  differentiating,  we  have, 

11  1  m  m 

x  =  (—2—"  =  an(z*  -  &)"*;  xm=  a"(z8-  Z>)~  » 


and, 


Substituting  in  (1),  and  reducing,  we  find, 

r  m    r  m     r    . 

n~  m 1  ^    —  T+S— 1 

ti 

(3) 

Which   can   be  integrated,  as  was  shown   above,    when 

m       r  . 

1-  -,  is  a  whole  number. 

n        s 

EXAMPLES. 

....  (4) 

ITl         4 

Here,  —  =  -  =  2 ;  hence,  the  second  criterion  is  satis- 
n       & 

fied. 

Comparing  (2)  with  (4),  we  find, 

a  =  1,  I  =  1,  n  =  2,  m  =  4,  r  =  1,  s  =  2,  and  z  =  (1  -f  x2)*, 

Hence, 

y*  /**  K        i 

/  (1  +  xz)*xsdx  =  I  (z9  —  l}zzdz  = 

«/  «/  53 


15 


METHODS   OF   INTEGKATIOK.  133 


=  dx(l  +  Z2)~        -*    .....  (5) 


Here,        -+_  —  __——  _  2  •  hence,  the  third  crite- 

n      s  <i      2 

rion  is  satisfied. 

Comparing  (3)  and  (5),  we  find, 

a  =  1,   5=1,   n  =  2,   m  =  —  3,    r  =  —  1,   s  =  2,   and 


_ 

FTence, 

dx 


Integration  by  Successive  Reduction. 

70.  When  an  integral  can  be  made  to  depend  upon  a 
simpler  integral  of  the  same  form,  it  is  evident  that  by 
successive  repetitions  of  the  process,  we  may  ultimately 
arrive  at  a  form  that  can  be  integrated  by  one  of  the  funda- 
mental formulas.  This  method  of  integration  is  called 
the  method  by  successive  reduction,  and  is  effected  by  for- 
mulas of  reduction,  sonic  of  which  it  is  now  proposed  to 
investigate. 

1°.  If  in  the  expression  for  a  binomial  differential,  we 
omit  the  factor  A,  represent  the  exponent  of  the  paren- 
thesis by  p,  and  factor  the  result,  we  have, 

fa  I-  b^xm"^dx  =  xm~n(a  +  bxn)W~*dx   .....   (1) 


134  INTEGRAL  CALCULUS. 

Let  u  =  xm~n,  and  dv  =  (a  +  fan)pxn~*dx. 

Differentiating  the  first,  and  integrating  the  second 
Formula  (29),  we  have, 

to  =  (m  -  n)x-^dx;  and  ,  =  ^ 


But,  (a  +  lxnf  +  l  =  (a  +  lxnf  (a  +  lxn)  ; 

a(g  +  te*)p       (a  +  &gn 
= 


Substituting  in  Formula  (28),  we  have, 


bn(p 


n(p 


Transposing  the  last  term  to  the  first  member,  and  re 
ducing,  we  have, 


ln(p  + 
Dividing  by  the  coefficient  in  the  first  member,  we  have, 


METHODS   OF   INTEGRATION.  135 

Formula  A  enables  us  to  reduce  the  exponent  of  the 
variable  without  the  parenthesis,  by  the  exponent  of  the 
variable  within  the  parenthesis,  at  each  application.  It 
fails  when  pn  +  m  =  0;  but  in  this  case,  the  third  crite- 
rion, (Art.  69),  is  satisfied,  and  the  expression  may  be 
integrated  by  a  previous  method. 

2°.  The  expression  (a  +  bxn)p  may  be  placed  under  the 
form, 


or,  a(a  +  lxnf~l  +  b(a  +  bxn)p~lxn; 
hence,  we  may  write  the  equation, 


f(a 
=  of  (a  +  fon)p-lxm-ldx  +  lj\a  + 


The  last  term  of  this  equation  can  be  reduced  by 
Formula  A.  Eeplacing  m  by  m  -f  n,  p  by  p  —  1,  and 
multiplying  by  Z>,  that  formula  becomes, 

if  (a  +  feY~1zm+"~1<*» 


pn  +  m          pn 
Substituting  in  (2),  and  uniting  similar  terms,  we  have, 

C  i)  m—\ 

j  (a  -*-  bx  yx   ~~  dx 
J  v 

(ct  -f-  bx    )    X  pnCt         f-  ,    7l\p 1    771—1  7  r  Tf\ 

:= + /  (#  ~r  C>iC   )          X          uX  • .  •  I  Jj\ 

pn  +  m         pn  +  m  «/ 


136  INTEGRAL   CALCULUS. 

Formula  B  enables  us  to  reduce  the  exponent  of  the 
parenthesis  by  1  at  each  application.  It  fails  in  the  same 
case  as  Formula  A.  When  m  and  p  are  negative,  we 
endeavor  to  increase  instead  of  diminishing  them.  For 
this  purpose  we  reverse  Formulas  A  and  B. 

3°.  Reversing  Formula  A,  and  reducing,  we  have, 

J\a  4-  lxn)pxm-n-ldx 

=  (a  +  ftQP*V»-»  _  »(,»  +  M)  f         lxnPx^dXt 
a(m  —  n)  a(m  —  n]  «/  v 

"Replacing  m  by  —  m  +  n,  we  have, 

f(a  +  WT.—1&  =  -  <fl  + 
./  v 

m  +  n)r        tofiyz 
v 


4°.  Reversing  Formula  B,  and  reducing,  we  have, 

f(a  +  lxnf-^xm-ldX 
=  _  (a  +  fa")V 


pna 
In  this,  substituting  —  p  +1  for^?,  we  have, 


J^ 

^ 


METHODS   OF   INTEGRATION.  137 

The  mode  of  applying  Formulas  A,  B,  C,  and  D,  will 
be  illustrated  by  practical  examples. 
1.  Let  it  be  required  to  integrate  the  expression, 


In  Formula  A,  making 

a  =.  r2,  b  —  —  1,  n  —  2,  p  =  £,  and  m  =  3, 
we  find, 

/'  i  /V2  _  ^2^7.      r2     /*  i 

(rt  _  <£*)*z*dx  =  -(—  -f-^  +  -^J  (r2  -  x*ydx 

.....  (3) 

In  Formula  B,  making 

a  =r2,  ft  =  —  1,  n  =  2,p  =  $,  and  m  =  1, 
we  find, 

*  ~ 


But 


/ON-i,         /       dx  .  —\x 

r*  —  xz)    ^dx  —  I  —  sin      -. 

«/  A/rg  _  ^2  r 


Substituting  this  in  (4),  and  that  result  in  (3),  we  have, 
Snally, 


/(r2  -  x*y 


4 
rz(rz  _  X2\^ 


138 

2.  Let  dy  = 


INTEGRAL   CALCULUS. 


=  (-  a*  +  x 


In  Formula  G,  making  a  =  —  a2,  b  =  1,  n  =  2,  p 
and  m  —  2,  we  have, 


/ 
-a 


But  by  Formula  (24), 

/_x-l  _i  7        /»      dk  1      _  i« 

—  a2  +  ic2)  %    1c?a;  =  I  —  -  --  =  -  sec     -. 
J  x^/xz  _  az      a  ,        a 

Substituting  in  (5),  we  have,  for  the  value  of  y, 


dx 
**- 


L_      __ 

SC 


3.  Let  dn  — 


In   Formula  D,  making  x  =  z,  a  =  1,  ft  =  1,   n  =  2, 

=  2,  and  m  =  1,  we  have, 


But, 


METHODS   OF   INTEGRATION.  139 

Hence,  we  have,  for  the  value  of  y, 


r      dz 

J    7*TT- 


h     tan-1* 

T     2  tf 


This  is  the  method  of  integration  referred  to  under  the 
Fourth  Case  of  rational  fractions.  By  a  continued  ap- 
plication of  Formula  D,  any  expression  of  the  form, 

7~^  --  2\n>  can  ^e  reduced  t°  an  integrable  form. 
(a  +  z  ) 

The  following  additional  examples  can  be  reduced  to 
integrable  forms  by  the  application  of  Formulas  A,  B,  C, 
and  D. 


_ 
sin      -. 

tl 


1  f/~  "if 

Ans.  y  =  -x(a*  —  x*)~2+  —sin"1-. 
A  6  a 


5.  dy  =  (a*  — 


6.  dy=(l+  x*)x*dx.       Ans.  y  =  - 


3    .    _ 


140  LNTEGKAL   CALCULUS. 


5°.  Let  it  be  required  to  simplify  the  expression, 
xndx 


Changing  the  form  and  removing  the  factor  x  from  under 
the  radical  sign,  we  have, 


in  formula  A,  making, 

a  =  2a,  b  =  -  1,  n  =  1,  p  =  —  J,  and  m  =  n  +  $, 
and  reducing,  we  have, 

l(pn  +  m)  =  —  n,  and  a(m  —  n)  =  2a(n  —  i)  =  a(2n—  1)  ; 
hence, 


71-1      , 

a;        V  2ax  — 


/*      XUdx 

I =   — 

•/   ^\/2ax  —  x2 

,9,,_1U/»    ,«-!& ^ 


Formula  E  enables  us  to  reduce  the  exponent  n  by  1, 
at  each  application  ;  if  n  is  entire  and  positive,  by  a  con- 
tinued application  of  the  formula  we  ultimately  arrive  at 

the  form,  /—     '        ,  which  is  equal  to  versin"1-. 
J  v2ax—x2  a 


1.  dy  = 


METHODS    OF    INTEGRATION.  141 

EXAMPLES. 

xdx 


J  /y 

Ans.  y  =  —  V%ax  —  x2  +  a  versin"  -. 
*  a 


Certain  Trinomial  Differentials. 

71.  Every  trinomial  differential  that  can  be  reduced  to 
the  form 

P 
(a  +  lx±  x*fxmdx (1) 

can  be  made  rational  in  terms  of  an  auxiliary  variable, 
and  consequently  can  be  integrated,  when  p  and  m  are 
whole  numbers. 

When  p  is  even,  the  expression  is  already  rational; 
when  p  is  odd,  say  of  the  form  2n  +  1,  the  expression 
becomes, 

(a  +  lx  ±  x*)n(a  +  lx±  x^xmdx (2) 

in  which  the  only  irrational  part  is  Va  4-  bx  =fc  x2.  There 
may  be  two  cases;  first,  when  xz  is  positive,  and  secondly, 
when  x2  is  negative. 


14:2  INTEGRAL   CALCULUS. 

1°.    When  xz  is  positive. 
Assume, 


Va  +  bx  +  x2  =  z  —  x;  /.  a  +  "bx  +  x2  =  z9  —  2zx  +  x*  j 
striking  out  the  common  term  x2,  and  solving,  we  have, 

z2  —  a 


;     /.  dx  = 

fyfi  T  0 

and 


4^4^ (3) 


Substituting  these  in  (2),  the  result  will  be  rational. 

EXAMPLES. 

~  (4) 


Comparing   (4)  with   (1),  we  find,  a  =  1,  and  5  =  1. 
Hence,  from  (3),  we  have, 


+  1 


Substituting  in  (4),  and   making  z  =  x  +  V 1  +  x  +  x2, 
we  hare, 


y  =  l(%z  +  1)  =  Z(2Z  +  1  +  2  VI 


^2  —  a;  —  1 


Ans.  y  =  Z(2a;  -  1  +  2Vx2  —  x  -  1). 


METHODS   OF  INTEGKATIOtf.  143 

3.rfy=__J|==. 

Ans.  y  —  if -====\. 

\          *     "^     I     ™    V  "^     r  *v     i     •**    X 

S°.   IFAew  x9  is  negative. 

Let  a  and  j3  be  taken  so  as  to  satisfy  the  equation, 

a  +  bx  -  x*  =  (x  -  a)(/J  -  x) (5) 

Assume 


Va  +  bx  —  x2  =  V(x  —  a)(#  —  x)  —  (x  —  a)g; 
squaring  and  reducing,  we  have, 

P-x  =  (x  —  a)z*  ;   or,  z  = 

hence, 

_  ff  +  ag8 

:T+"^"; 

and 


These  values  substituted  in  (1),  will  make  it  rational. 


T   ,    , 

Let  e     = 


EXAMPLE. 

dx  dx 


-*) 

.-.  a  =  -  2,  and  /S  =  1 (7) 


144  INTEGRAL   CALCULUS. 

From  (C),  we  have, 
6zdz 


Substituting  in  (7),  we  find, 


also  z  =  A  /  - — — . 

A 


=  -  2tan-1 


r+  zz '  •  •  y  -  |/  jr+2 


Integration  by  Series. 

72.  It  is  often  convenient  to  develop  a  differential  into 
a  series,  arranged  according  to  the  ascending  powers  of 
the  variable,  and  then  to  integrate  each  term  separately. 
This  method  sometimes  enables  us  to  find  an  approxi- 
mate value  for  an  integral,  which  cannot  be  found  in  any 
other  manner;  it  also  enables  us  to  deduce  many  useful 
formulas. 

EXAMPLES. 

1.  dy  = 


Developing  (1  —  x2)  *    by   the  binomial  formula,  and 
multiplying  each  term  by  x2dx,  wre  have, 


X3          X6         X 


•'•y=w  -TO  -56-ifl  -eta 


METHODS   OF   INTEGRATION.  145 

By  Formula  (14)  y  is  equal  to  tan"1^.     Developing  bj 
the  binomial  formula,  we  have, 

(1  +  x*)~l  =  1  —  x2  +  ?;4  —  x*  +  x8  —  z10  +  etc. 
Substituting,  and  integrating,  we  have  the  Formula, 

__1  x3      x5       x'1       x9 

tall     z=a,.__  +  ___  +  __etc. 


=  (1  —  x  4-  xz  —  x*  +  etc.)dx 
y  =  1(1  +  x)  =  x  -  |-  +  f-  -  |-  +  etc. 


3xs 


Integration  of  Transcendental  Differentials. 

73.  A  transcendental  differential  is  one  that  is  expressed 
in  terms  of  some  transcendental  quantity.    There  are  three 
principal  classes;    viz.,  logarithmic,  exponential,  and  cir- 
cular. 

Logarithmic  Differentials. 

74.  A  large   ir'imber  of  cases  come  under  the  general 
form, 

dy  =  zm-l(lx)ndx (1) 


146  INTEGRAL   CALCULUS. 

Assume, 

u  =  (Ix)71,  and  dv  =  xm~ldx; 

.:  du  =  — ^-^ '  -'  and  v  =  —  > 

2;  T/Z. 

substituting  in  Formula  (28),  and  reducing,  we  have. 


y  = 


Formula  ,F  is  a  formula  of  reduction  ;  it  reduces  the 
exponent  of  (Ix)  by  1  at  each  application. 

EXAMPLES. 
1.  dy  =  x(to)*dx. 


2.  rfy  = 

3.  dti  = 


Am.  y  = 
Eeversing  Formula  ^  and  reducing,  we  have, 


Replacing  w  —  1  by  —  w,  and  reducing,  we  have, 

/^J_^  =  __  x^  __  +      m       /?^~V* 
(lx}n  (n  -  1)  (Ix)71'1  +  n-  lJ\U)>1-1 


METHODS   OF   INTEGRATION.  147 

The  continued  application  of  formula  G  gives  as  a  final 

/vm— lflx 
n  .—,  which  can  be  integrated 
(ix) 

by  series.     It  fails  when  n  =  1.     Let  m  =  0,  and  n  =  I ; 
we  then  have, 


Exponential  Differentials. 

75.  Let  it  be  required  to  integrate  an  expression  of  the 
form, 

dij  =  xmaxdx (1) 

Assume, 


u  —  xm,  and  dv  —  axdx\   .:  du  =  mxm     dx,  and  v  =  r— . 

la 


0* 

u  =  mx""    ~tlx,  and  v  = 
Substituting  in  Formula  (28), 

«/  la         la  J 

When  m  <  1,  we  have, 

fifdx  =  <F  la      fifdx 

**  xm   ~         (m-l}xm-1      m-l^x™-1 


INTEGRAL   CALCULUb. 


EXAMPLES. 


1.  dy  =  xaxda\  A  us.  .y  —  ~j~\x  ~~  T~  )' 

2.  dy  =  x*axdx. 

axf  3       3«2         (jx  6    \ 

AnS.y   =   T-\X       --  :  --  \-    yy-r-    —  I. 

te\          /«       (/rt)2      (/«)  v 

e'Yfa  ^       -/e^fo 

3.  dy  —  -  j-  .  Ans.  y—  --  +  I- 

x2  x     J    x 

But  by  means  of  McLaurin's  Formula,  we  find, 


hence, 

x       ,        xdx 


which,  being  substituted  above,  gives, 


4.  dy  =  x*exdx.  Ans.  y  =  ex(x*  -  Zx  +  2). 


Circular  Differentials. 

76.  Circular  differentials  may  be  integrated  by  the 
methods  of  transformation  and  successive  reduction,  or 
they  may  be  reduced  to  algebraic  forms  by  making 


METHODS   OF   INTEGRATION.  149 


EXAMPLES. 


-    T  ,   ,         dx 

1.  Let  av  =  -  —  . 

smx 


Prom  trigonometry,  we  have, 

smx  =  2sin^cosjx  =  2tanJ#cos* 


or, 

dx  dx 


smx  ~  2tan  %xcos2^x  ~      tanjz  " 
Hence, 


-  =  Z(tanjz)  .....  (1) 
smx 


0   T   ,    7 

2.  Let  dy  =  -  . 
cosx 

If  we  make  x  equal  to  90°  —  x  in  the  preceding  formula, 
and  reduce,  we  have, 


3.  Let  dr    = 


From  trigonometry,  we  have, 


Hence, 


sinz  do?        ^Ircosa; 

tana;  =  --  ;  or,  -  -  =  —  ;  ---  =  —  ^ 

coso:          taiue         smx          smx 


=Z(sina;)  .....  (3) 
tuna; 


150  INTEGRAL   CALCULUS. 

I        T      1       7  $X 

4.  Let  ay  =  —  —  . 

cote 

Making  x  equal  to  90°  —  x  in  (3),  and  reducing,  we  have, 

rdx  =  _         .....  ( 

J  cotx 

5.  Let  dy  =  -J^—. 

smftoosz 

From  trigonometry,  we  have, 

sin^cosa;  = 

we  also  have, 

dx  = 

hence, 

dx       _d(2x) 


Applying  Formula  (1),  we  have, 

r  dx 

J  s 


Let  us  have  the  expression, 

dy  —  sinmo;cosn^^  .....  (6) 
Making  sinz  =  z,  whence  cosx  —  *J\  —  zz, 


and  dx  =  —  —  -  —  . 
Vl  -  z* 
we  have, 

71-1 

dy  =  zm(l  -  z2)    3  dz  .....  (7) 

Form  (7)  can  always  be  integrated  when  m  and  n  are 
whole  numbers,  because  it  will  then  satisfy  one  of  the  three 
criterions  (Art.  09). 


METHODS   OF   INTEGRATION.  151 

6.  dy  —  &'mzxcos&xdx. 

Comparing  with  (6).  w*  find  in  =  2.  and  n  --  3;  hence, 
from  (7),  we  have, 


dy  =  (1  -  z*)z*dz;     /.  y  =    -  -  ~  = 

•  i  0 

7.  Let  dy  =  sh\*xdx. 


Here  m  =  3,  and   n  —  0;   and   dy  =  (1  —  z*) 
making  x  =  z,  a  =  1,  £  =  —  1,  n  =  2,  j9  =  —  J,  and  m  =  4, 
in  Formula  A,  we  have, 


But  from  Formula  (29), 


hence, 


\  (1  -  zrfz*  -  |(1  -  z«)t  ='  -  i 


8.  Let  dy  =  coszxdx  =  dz^/\  —~z2. 
Applying  Formula  (B),  we  find, 


But, 


152  INTEGRAL   CALCULUS. 

Hence, 

y  —  -(coszsince  +  x). 

^ 

flnft  (1  y, 

9.  Let  dy  =  ^^  =  ^— =  (1  -  z8) 

Applying  Formula  (7,  we  have, 


But, 


Hence, 


10.  Let  c?v  =  -,--%-  =    ,.,   ^  ,,,  =  (!-**)"  V«£fe, 

22 


By  Formula  Z),  we  have, 

/(I  _  g«) 
But, 


sin.rcosa; 
Hence, 

?/  =  —  i—  4-  /(tana;). 
4/       2cos2o; 


METHODS   OF   INTEGRATION.  153 


11.  Let  dy  =  tsm^xdx  =  (1  —  z2)' 
y  —  Jtanso;  —  tanjc  4-  x. 

EXAMPLES   IN   SUCCESSIVE   INTEGRATION. 

Making  sinz  =  z,  whence,  dzy  =  z(dz)2,  we  have, 
dy      z2  z3 

&  =  a"  +    '       y  =  6~  +      +    ' 

or, 

=  ^£_?  +  ^g^^,  +  @m 
o 

13.  c?2y  =  cosxsm*xdx2  =  cosa:(</coscc) 8 ; 

^?y          1 

**'   /7f  \  =  o  COS  x  +   ^« 

and 

y  =  -  co&3x  +  Cfcosa;  +  6*. 
b 


Integration  of  Differential  Functions  of  two  Variables. 

77.  A  total  differential  of  a  function  of  two  variables  is 
of  the  form, 

<&  =  Pdx  +  Qdij (1) 

7* 


154  LNTECltAL   CALCULUS. 

In  which  P  and  Q  are  functions  of  x  and  y ;  but  differ- 
ential expressions  of  that  form  are  not  necessarily  total 
differentials.  That  they  may  be  so,  they  must  (Eq.  5, 
Art.  2G),  satisfy  the  condition, 

dP      dQ 

dy  ^  dx~ W 

When  an  expression  of  the  form  (1)  is  to  be  integrated, 
we  first  apply  the  test  expressed  by  (2) ;  if  that  be  satisfied, 
the  expression  is  integrable.  To  perform  the  integration, 
integrate  one  of  the  partial  differentials  with  reference  to 
its  corresponding  variable,  that  is,  as  though  the  other 
variable  were  constant,  and  add  such  a  function  of  that  va- 
riable as  will  satisfy  equation  (1).  Thus,  to  integrate  ex- 
pression (1),  we  have, 

z  =  Cpdx  +  R (3) 


in  which  R  is  a  function  of  y  alone.  The  value  of  R  may 
be  found  by  differentiating  the  second  member  of  (3)  with 
respect  to  y,  and  placing  its  partial  differential  coefficient 
equal  to  Q.  Hence,  we  have, 


d/Pdx 


dy          dy 

Transposing,  multiplying  by  dy,  and  integrating  with 
respect  to  y,  that  is,  as  though  x  were  constant,  we  find, 


METHODS   OF   INTEGRATION.  155 

Substituting  in  (3),  we  find, 


//Y         dfpdx 
Pdx+J  (Q- 


EXAMPLES. 

1.  dz  =  3xzy2dx  +  2x*ydy. 

Here  P  =  3x*ij*,   and  Q  =  2x*y,  and  (2)  is  satisfied. 
Integrating  by  Formula  (4),  we  hare, 


z  =  z3y2  +      2x9y  —  2x*y)dy  =  x*y*  -f  C. 

2.  dz  =  ydx  +  xdy. 
The  test  (2)  is  satisfied.    Integrating  by  (4),  we  have, 

z  =  yx  +  i  (xdy  —  xdy)dy  —  yx  -f  0. 


By  Formula  (4),  we  have, 


z  = 


4.  dz  =  (oxy  —  y*)dx  +  (3a;2  —  %xy}dy 
+ 

—  y*x  +  f(3xz  —  2xy  —  3x9 


PART  IV. 

APPLICATIONS  OF  THE  INTEGRAL  CALCULUS. 


I.  LENGTHS  OF  PLANE  CURVES. 

Rectification. 

78.  Rectification  is  the  operation  of  finding  an  expres- 
sion for  the  length  of  a  curve.  This  may  be  done  by 
finding  an  element  of  the  length,  as  explained  in  Art.  51, 
and  then  integrating  it  between  proper  limits.  If  we 
apply  the  integral  sign  to  Equation  (1),  Art.  51,  we  have, 


L  =Vdx*  +  dy*  .....  (1) 

1°.  To  rectify  the  semi-cubic  parabola,  whose  equation  is, 


Differentiating  and  substituting  in  (1),  we  have, 

(2) 

Integrating  by  Formula  (29)  ,  we  have, 

L  =  ^(4a*  +  9y)»  +  0  .....  (3) 
As  explained  in  Art.  GO,  this  integral  is  indefinite,  anti 


LENGTHS   OF   PLANE   CURVES.  157 

expresses  the  length  of  an  arc  from  any  ordinate  up  to  tiny 
other  ordinate. 

If  we  estimate  the  length  from  a  point  whose  ordinate 
is  0,  the  length  at  that  point  is  0,  and,  from  (3),  making 
Ly  and  y  —  0,  we  have, 


Substituting  this  value  of  C  in  (3),  and  denoting  the  cor- 
responding value  of  L  by  L',  we  have, 


This  is  a  particular  integral,  and  it  expresses  the  length 
of  the  arc  from  the  particular  ordinate  0,  up  to  any 
ordinate  y. 

If  we  wish  to  find  the  length  from  the  particular  ordi- 
nate 0,  up  to  an  ordinate  #,  we  make  y  —  b,  in  (4).  Doing 
so,  and  denoting  the  corresponding  value  of  L'  by  JJ\ 
we  have, 

"=       ••  +  »)*-  IT  .....  (g) 


This  is  the  definite  integral,  and  expresses  the  length  of 
the  arc  from  the  ordinate  0,  up  to  the  ordinate  b. 

In  this  case  the  limits  are  0  and  I  ;  the  integral  may  be 
found  otherwise,  as  follows  : 

Making  y  =  0,  in  (3),  and  denote  the  corresponding 
value  of  L  by  LQ,  we  have, 


158  INTEGRAL   CALCULUS. 

Making  y  =  #,  in  (3),  and  denoting  the  corresponding 
value  of  L  by  Lb,  we  have, 

X4  =  ^(4«t  +  9S)*  +0  ----  (7) 

Equation  (6)  expresses  the  length  of  the  curve  from  any 
point  up  to  the  point  whose  ordinate  is  0,  and  (7)  ex- 
presses the  length  of  the  arc  from  the  same  point  up  to  a 
point  whose  ordinate  is  b  ;  the  first  taken  from  the  second 
will  therefore  be  the  definite  integral,  and  will  express,  as 
before,  the  length  of  the  arc  from  the  point  whose  ordinate 
is  0,  up  to  the  point  whose  ordinate  is  b.  Making  the 
subtraction,  and  adopting  the  notation  explained  in  Art.  60, 
we  have, 

b 


L"  = 

0 

The  same  result  as  found  in  (5).    In  this  case  the  initial 

3 

abscissa  is  0  and  the  terminal  abscissa  is  -  . 

a 

2°.  To  rectify  the  cycloid,  wJiose  differential  equation 
(Art.  56),  is 


-  y 
Substituting,  in  (1),  and  reducing,  we  have, 


L  = 


Integrating  between  the  limits  0  and  2r  Formula  (29), 
we  have, 


2r 


LENGTHS   OF   PLAHE   CURVES.  159 

This  is  the  expression  for  half  of  one  branch.  Hence,  the 
whole  branch  is  equal  to  eight  times  the  radius  of  the 
generating  circle. 

3°.  To  rectify  the  circle,  whose  equation  is, 

y  =  (r*  -  xrf. 

Differentiating,  substituting  in  (1),  and  developing  by 
the  binomial  formula,  we  have, 

L  =  rf(r*  -  x*)~tdx  =  f(dx  +  ^  .  x*dx  +  ^ 


Performing  the  integration,  making  r  =  1,  and  com- 
mencing the  arc  at  the  point  whose  abscissa  is  0,  that  is, 
at  the  upper  extremity  of  the  vertical  diameter,  we  have, 


Formula  (13)  may  be  used  for  finding  the  length  of  an 
arc,  but  the  series  is  not  very  converging,  and  therefore 
the  process  is  tedious.  We  know,  when  x  =  J,  that  L'  is 
equal  to  an  arc  of  30°,  that  is,  to  -J-tf.  Making  L'  =  $*, 
and  x  =  £,  we  have, 

5*  =  I  +  sis  +  ^75  +  W&J  +  etc"  =  - 


=  3.1415 


4°.  A  better  method  for  deducing  the  value  of  if,  is  to 
find  the  length  of  the  arc,  in  terms  of  its  tangent. 

If  L  =  tan     a;,  we  have  (Art.  18), 

dL  =  =  1  +  x'~1<lx- 


160  INTEGRAL   CALCULUS. 

Developing  by  the  binomial  formula,  we  have, 

dL  =  dx(l  -  x*  -f  x*  -  x6  +  x8  -  etc.)  .....  (15) 
Integrating,  and  determining  (7,  as  before,  we  have, 
x3      xs      x11      x9 

i=*-T+5--r  +  --etc 


To   render    (16)    converging,   x  must  be   made   small 
Assume  the  trigonometrical  formula, 

tana  +  tanJ 


and  let  a  —  tan"1???,  b  =  tan"1^,  and  a  +  I  =  tan    l«j 

then  will 

m  +  n  z  —  m  nff. 

z  —  -        -  ;  or,  n  —  —    -=•  .....  (17) 
1  —  mn  mz  +  1 

1  2 

[f  z  —    1,  and  m  =  -,  we  have,  n  —-\ 
o  o 

/.  tan"1  1  =  tan"1^  +  tan"1! 

0  o 

21  7 

If  z  =    -,  and  wi  =  -,  we  have,  n  —  ^  ; 


71  9 

If  2  =  —  ,  and  m  =  -,  we  have,  w  =  —  ; 

17  0  4:0 


. 


91  1 

If  z  =  —  ,  and  OT  =  -,  we  hare,  n  -    -  ^ 


AREAS   OF   PLANE   CURVES.  1C] 

Hence,  "by  successive  substitution,  we  find, 

-4  =  tan"1  1  =  4tan-1i  -  tan"1-^-  .  .  .\.  .  (18) 
4  5  /coJ 


If  we  make  x  —  -,  in  Equation  (1C),  we  find  the  value  of 
5 

tau—1r,  and.  in  like  manner,  we  find  the  tan      ^r;  sub- 

0  /vO  J 

stituting  them  in  (18),  and  reducing,  we  have  the  value  of 
if.  With  very  little  labor,  the  value  of  if  may  be  found  to 
8  or  10  places  of  decimals. 


II.  AREAS  OF  PL  AXE  CURVES. 

Quadrature. 

79.  Quadrature  is  the  operation  of  finding  an  expression 
for  the  area  of  a  portion  of  a  plane  bounded  by  a  curve, 
the  axis  of  abscissas,  and  any  two  ordinates.  This  is  done 
by  finding  an  expression  for  the  elementary  area,  as  ex- 
plained in  Art.  51,  and  then  integrating  the  result  between 
proper  limits.  Applying  the  sign  of  integration  to  For- 
mula (2),  Art.  51,  we  have, 


A  = 


1".  To  find  the  area  of  a  parabola,  whose  equation  is, 

2/2  —  2px. 
Finding  the  value  of  ?/,  and  substituting  in  (1),  we  have, 


A  =  Vtyxdx  =  .  a    +  0  .....  (2) 


162  INTEGRAL   CALCULUS. 

If  we  commence  the  area  at  the  ordinate  0,  we  have,, 
C  =  0  ;  hence, 


(3) 


That  is,  the  area  of  any  portion  of  a  parabola,  reckoned 
from  the  vertex,  is  two-thirds  the  rectangle  of  its  terminal 
co-ordinates. 

2°.  To  find  the  area  of  a  circle,  whose  equation  is9 


y  -  V 
tf  ibstituting  in  (1),  we  find, 


A 


=  C(r*  -xrfdx  .....  (4) 


Reducing  by  Formula  J3,  and  integrating  the  last  term 
by  Formula  (20),  we  have, 

_*(r»-*')* 


Taking  the  integral  uetween  the  limits,  x  =  0  and  x  =  r, 
which  gives  the  area  of  a  quadrant,  and  remembering  that 

sin""1 1  —  sin""1  0  =  Jtf,  we  have, 

T"  1  1  cfT*2 

/I"  =  —[sin      1  —  sin      0]  =  -: — ;  .'.  Area  =  *7'2  . .  .  (6) 

A  4: 

3°.  To  find  the  area  of  the  ellipse,  whose  equation  is, 

«/  r-_//rt   _  ^2\i 


AREAS   OF   PLANE   CURVES.  163 

Substituting  in  (1),  and  integrating,  as  in  the  last  example, 
we  have, 


Integrating  from  x  =  0  to  x  =  a,  and  multiplying  by  4,  we 
Lave,  for  the  area  of  the  ellipse, 

±A"  =*ab (8) 

4°.  To  find  the  area  of  the  cycloid,  whose  diffeimtial 
equation  is, 


Substituting  in  (1),  we  find, 

f*    y*dy 

—  f  —        — 


Applying  Formula  E   twice,  and  Formula  (26)  once,  wo. 
find, 


yV%ry  -  y* 
~2~ 

+  <r  ~  V%ry  —  y*  +  rversin"1  ^1  4-  0. 

Taking  the  integral  between  the  limits  y  =  0,  and  y  =  2r, 
and  multiplying  by  2,  we  find,  for  the  area  of  the  cycloid, 


2A"  =  3«r*  .....  (10) 

That  is,  the  area  between  one  branch  and  the  directing 
line  is  three  times  the  area  of  the  generating  circle. 

5°.  To  find  the  area   of  the  logarithmic  curve,   whose 
equation  is,  y  =  Ix. 


164  INTEGRAL   CALCULUS. 

Substituting  iu  (1),  we  have, 

A  =f(lx)dx  ----  (11) 

Making  m  =  1  and  n  —  1,  in  Formula  F,  and  reducing, 
we  have, 

A  =  xlx  -  x  +  0  .....  (12) 

If  we  commence  the  arc  at  the  ordinate  corresponding  to 
<r.  =  1,  we  have,  (7=1,  and 

A'  =  x(to-l)  +  l  .....  (13) 

6°.  To  find  the  area  of  a  rectangular  hyperbola^,  bounded 
by  the  curve,  one  asymptote,  and  any  two  ordinates  to  that 
asymptote. 

Assume  the  equation, 

xy  =  m,  and  make  m  =  1,  whence  y  —  -. 

x 

Substituting  in  (1),  we  have, 

A  =      =  lx  +  c 


Commencing  the  area  from  the  ordinate  through  the  vertex, 
where  x  =  1,  we  have,  (7=0;  hence, 

A'=lx  .....  (15) 

That  is,  the  area  commencing  from  the  ordinate  through 
the  vertex  is  the  Napierian  logarithm  of  the  terminal  ab- 
scissa. Had  we  not  made  m  —  1,  we  should,  in  like  man- 
ner, have  found 

A'  =  mlx. 

In  which  the  area  is  equal  to  the  logarithm  in  a  system 
whose  modulus  is  m. 


AREAS  OF  SURFACES  OF  REVOLUTION.       IG5 


III.   AREAS  OF  SURFACES  OF  REVOLUTION. 

Surfaces  Generated  by  the  Revolution  of  Plane  Curves. 

8(fc  The  area  of  a  portion  of  a  surface  of  revolution, 
bounded  by  two  planes  perpendicular  to  its  axis,  is  de- 
termined by  finding  an  expression  for  an  elementary  zone, 
as  explained  in  Art.  51,  and  integrating  the  result  between 
proper  limits.  Applying  the  sign  of  integration  to  For- 
mula (3),  Art.  51,  we  have, 


+  dy*  .....  (1) 

1°.  To  find  the  surface  of  a  sphere,  the  equation  of  the 
generating  circle  being,  y  •=.  (r2  —  x2)^. 

Differentiating,  we  have,  dy  —  —  (r2  —  x2)~ixdx;  sub- 
stituting the  values  of  y  and  dy  in  (1),  and  integrating 
from  —  r  to  +  r,  we  have, 


=  2«rjdx  =  4«r* (2) 


8" 

—  r 

Hence,  the  area  of  the  surface  of  a  sphere  is  equal  to  four 
great  circles,  or  to  two-thirds  the  surface  of  the  circum- 
scribed cylinder. 

2°.  To  find  the  surface  of  a  right  cone. 

The  equation  of  the  generating  line,  the  vertex  of  the 
cone  being  at  the  origin,  is 

y  =  ax  ;    /.  dy  =  adx. 


1.66  INTEGRAL   CALCULUS. 

Substituting  in  (1),  and  reducing,  we  have, 

S  =  ZiraVl  +  azjxdx  =  «ax2  Vl  +  a2  +  C  .....  (3) 

If  the  initial  plane  pass  through  the  vertex,  we  have,  for 
that  plane,  S  =  0,  and  x  =  0,  whence  (7=0;  hence, 

Sr  =  *ax2Vl  +  a2  =  *y  X  #A/1  +  a2  .....  (4) 

But  a  is  the  tangent  of  the  semi-angle  of  the  cone,  and 
consequently  x^\  +  a2  is  the  slant  height;  2*y  is  the 
circumference  of  the  cone's  base;  hence,  the  convex  sur- 
face is  equal  to  half  the  circumference  of  the  base  into  the 
slant  height. 

3°.  To  find  the  surface  of  the  paraboloid  of  revolution. 
Assume  the  equation  y2  =  2px,  whence, 


y  =      %px,  and  dy2  —  —  dx2. 

&X 

Substituting  in  (1),  and  reducing,  we  have, 


a  =  2*  fip*  +  2px)ldx  =    -  (p2  +  2px)?  +  C  ----    (5) 
If  the  initial  plane  pass  through  the  vertex,  we  have, 


Which,  in  (5),  gives, 


4C.  To  find  the  surface  generated  by  revolving  one  branch 
of  a  cycloid  about  its  base. 


VOLUMES   OF   SOLIDS   OF    REVOLUTION.  1G7 

Assuming  the  equation,  dx  =  db  —  —  -,  and  substi- 


tuting  in  (1),  we  have, 

(7) 


Applying  Formula  A,  and  integrating  the  last  term  by 
Formula  (29),  we  have, 


(2r  -  y)     +       (2r  -  y)       +  6'. 
Integrating  between  the  limits  y  —  0  and  ?/  =  2r,  we  havo, 


-r»  .....  (8) 


This  is  the  surface  generated  by  half  of  one  branch. 
Hence,  to  find  the  whole  surface,  we  multiply  by  2.  This 
gives 


IV.  VOLUMES  OF  SOLIDS  OF  REVOLUTION. 

Cubature. 

81.  Cubature  is  the  operation  of  finding  the  volume  of  a 
solid.  When  this  is  bounded  by  a  surface  of  revolution 
and  two  planes  perpendicular  to  its  axis,  the  volume  may 
be  ascertained  by  finding  the  volume  of  an  element 
bounded  by  two  such  planes  infinitely  near  to  each 
other,  as  explained  in  Art.  51,  and  then  integrating  the 


108  INTEGRAL   CALCULUS. 

result  between  proper  limits.     Applying  the  sign  of  inte- 
gration to  Formula  (4),  Art.  51,  we  have, 


V  =  *fy*dx (1) 


1°.   To  find  the  volume  of  a  sphere. 

Assume  the  equation,  yz  —  r2  —  x2,  and  combine  it  with 
(])  ;  we  have, 

C. 

Taking   the   integral   between   the  limits,  x  =  —  r,   and 
x  =  +  r,  we  find, 

V"  =  «2r*  ~~    =    T»  =  4*r2  X    r  .....  (2) 


That  is,  the  volume  is  equal  to  the  surface  ~by  one-third  the 
radius. 

2°.  To  find  the  volume  of  a  spheroid  of  revolution. 

There  are  two  species  of  spheroids  of  revolution. 

1st.  The  prolate  spheroid,  generated  by  revolving  an 
ellipse  about  its  transverse  axis. 

2dly.  The  oblate  spheroid,  generated  by  revolving  an 
ellipse  about  its  conjugate  axis. 

1st.  The  prolate  spheroid.     In  this  case  the  equation  of 

£2 

the   meridian   curve  is,   y2  =  —  (a2  —  xz).    Substituting 

this  in  (1),  and  integrating  between  the  limits  x  =  —  a, 
and  x  =  +  a,  we  have, 


-  x*)dx  =  z«b*a  =  !*£*  X  2a  .....  (3) 


VOLUMES   OF   SOLIDS   OF   KEYOLUTIOJS".  1G9 

That  is,  the  volume  is  equal  to  two-thirds  of  the  circum- 
scribing  cylinder. 

2dly.   The  oblate  spheroid.     In  this  case,  if  the  conjugate 
axis  coincide  with  the  axis  of  x,  the  equation  of  the  meri- 

a2 
dian  curve  is,  y2  =  —  (b2  —  x2).     Substituting  in  (1),  and 

integrating  from  —  b  to  +  I,  we  have, 


V"  =  \«a2b  =  l«a2  X%b  .....  (4) 
o  o 

Hence,  as  before,  we  have,  the  volume  equal  to  two-thirds 
the  circumscribing  cylinder.^ 

In  both  cases,  if  a  =  b  —  r,  we  have, 


V"  =    ^3  .....  (5) 

This  hypothesis  causes  the  ellipsoids  to  merge  into  tne 
sphere. 

3°.  To  find  the  volume  of  a  paraboloid  of  revolution. 

The  equation  of  the  meridian  curve  is,  y2  =  2px.    Hence, 
from  (1),  we  have, 

V  =  %«pixdx  =  «px2  +  C  .....  (6) 

If  the  initial  plane  pass   through  the   vertex,  we  have 
(7=0,  and 

V  =  «px2  =  «y2  X  \x  .  .     .  .  (7) 

That  is,  the  volume  is  equal  to  half  the  cylinder  that  has  the 
same  base  and  the  same  altitude. 


170  INTEGRAL  CALCULUS. 

4°.  To  find  the  volume  generated  by  revolving  one  branch 
of  the  cycloid  about  its  base. 

The  differential  equation  of  the  meridian  curve  is, 


hence,  from  (1),  we  have, 


Keducing  by  Formula  E,  integrating  the  last  term  by  For- 
mula (26),  and  taking  the  integral  between  the  limits, 
y  =  0,  and  y  —  2r,  we  find,'  for  one-half  the  volume 
required, 


V"  =       *r3,  or2F"  =  5*2/-3  =flr(2r)    X  2«r  .....  (8) 

A  o 

Hence,  the  volume  is  equal  tofive-eighklis  the  circumscribing 
cylinder. 

5°.  To  find  the  volume  generated  by  revolving  the  loj'triih- 
mic  curve  about  the  axis  of  numbers. 

The  equation  of  the  generatrix  is  y  =  Ix.     Hence, 
V=  «C(lxydx  .....  (9) 


Reducing  by  Formula  F,  we  have, 

V  =  «[x(lx)*  -  2(xlx  -  x)]  +  0. 

If  the   initial   plane    pass   through   the   point    whose 
abscissa  is  1,  we  have,  C  —  —  2^.     Hence, 

2+1)]  .....  (10) 


PART  V. 

APPLICATIONS  OF  THE  DIFFERENTIAL  AND 
INTEGRAL  CALCULUS  TO  MECHANICS  AND 
ASTRONOMY. 

I.  CENTRE  OF  GRAVITY. 

82.  In  what  follows,  bodies  are  supposed  to  be  homoge- 
neous ;  the  weight  of  any  part  of  a  body  is  therefore  pro- 
portional to  its  volume,  and  consequently  the  weight  of 
the  unit  of  volume  may  be  taken  as  the  unit  of  weight. 
Points,  lines,  and  surfaces  are  supposed  to  be  material: 
A  material  point  is  a  body  whose  length,  breadth,  and 
thickness  are  infinitesimal ;  a  material  line  is  a  line 
whose  length  is  finite,  and  whose  breadth  and  thickness 
are  infinitesimal;  a  material  surface  is  a  body  whose 
length  and  breadth  are  finite,  and  whose  thickness  is  infi- 
nitesimal; under  this  supposition  a  point  is  an  elementary 
portion  of  a  line,  a  line  is  an  elementary  portion  of  a  sur- 
face, and  a  surface  is  an  elementary  portion  of  a  solid. 

The  weights  of  the  elements  of  a  body  are  directed 
toward  the  centre  of  the  earth,  and  because  the  bodies 
treated  of  are  exceedingly  small  in  comparison  with  the 
earth,  these  weights  may  be  regarded  as  a  system  of  par- 
allel forces;  hence,  the  weight  of  a  body  is  equal  to  the 
sum  of  the  weights  of  its  elements  and  is  parallel  to 
them. 


172  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

The  centre  of  gravity  of  a  body  is  a  point  through  which 
its  weight  always  passes.  This  point  may  be  found  by  the 
principle  of  moments,  which  may  be  enunciated  as  follows: 
the  moment  of  the  resultant  of  any  number  of  forces,  with 
respect  to  an  axis,  is  equal  to  the  algebraic  sum  of  the 
moments  of  the  forces,  with  respect  to  the  same  axis. 
(Mechanics.,  Art.  35.) 

In  applying  this  principle,  we  assume  the  following  re- 
sults of  demonstrations  in  mechanics  :  1°.  The  centre  of 
gravity  of  a  straight  line  is  at  its  middle  point;  2°.  The 
centre  of  gravity  of  a  plane  figure  is  in  that  plane  ;  3°.  If 
a  plane  figure  have  a  line  of  symmetry,  its  centre  of  gravity 
is  on  that  line  ;  and  4°.  If  a  solid  have  a  plane  of  symmetry, 
its  centre  of  gravity  is  in  that  plane.  (Mechanics,  Arts.  44, 
45,  46.) 

To  deduce  general  formulas  for  finding  the  centre  of 
gravity  of  a  body,  assume  a  system  of  co-ordinate  axes  that 
are  to  retain  a  fixed  position  with  respect  to  the  body,  but 
that  change  position  when  the  body  moves;  denote  the 
volume,  and  consequently  the  weight,  of  any  element  of 
the  body  by  dv,  and  the  co-ordinates  of  its  centre  of 
gravity  by  x,  y,  and  z;  denote  the  weight  of  the  body  by 

v  =  I  dv,  and  the  co-ordinates  of  its  centre  of  gravity  by 
a,,  y1?  and  zr 

If  the  body  be  placed  in  such  a  position  that  the  plane 
xy  is  horizontal,  the  weights  of  the  elements  and  of  the 
body  are  parallel  to  the  axis  of  z,  the  moment  of  the  body, 


with   respect  to  the   axis   of  y,   is   ^1  dv,  the   moment 
of  any  element,  with   respect   to   the   same  axis,  is 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      173 

and  the  algebraic  sum  of  the  moments  of  all  the  elements 
\alxdvj  hence,  from  the  principle  of  moments,  we  have, 

/„                         fxdv 
lv  =  Ixdv  ;     .:  x1  =  -~ — (1) 
J                               Idv 

In  like  manner,  we  have, 


If  the  body  be  turned  about  so  that   the  plane  yz   is 
horizontal,  we  have,  in  like  manner, 


CdV=rydv;    s.y  !=?-%- (2) 

»/          «/  I  dv 


//* 
!v=lz 
J 


I  zdv 
zdv;     .:  zl  =J7— (3) 

/* 


"When  the  body  is  in  a  plane,  that  plane  may  be  taken 
as  the  plane  xy,  in  which  case  zl  =  0  ;  if  the  body  have  an 
axis  of  symmetry,  that  may  be  taken  as  the  axis  of  x,  in 
which  case  z1  =  0,  and  yl  —  0. 

Centre  of  Gravity  of  a  Circular  Arc. 

83.  Let  the  radius  perpendicular  to  the  chord  of  the  arc 
be  taken  as  the  axis  of  x ;  then  will  zlt  and  y}  be  equal  fc 
0.  Denote  the  radius  of  the  circle  by  r,  ^  A 
the  chord  by  cy  and  the  arc  by  A.  The 
origin  being  at  the  centre,  the  equation  of 
the  arc  is,  y2  =  r2  —  x2 ;  hence, 

th  = 


=-TC 


c 

Fig.  16. 

rdy  __        rdy 


x 
12* 


174  DIFFERENTIAL   AND    INTEGRAL   CALCULUS. 

Substituting  in  (1),  and  integrating  between  the  limits, 
y  =  —  j£,  and  y  =  +  \c,  we  "have,  for  the  numerator, 


Jrdy  =  rc, 


and  for  the  denominator, 


//.2_2/s         L  2r  2rJ 

-ic 

Hence, 


/*•/» 

:  :  r  :  a: 


That  is,  /Ae  ce?^re  of  gravity  of  the  arc  of  a  circle  is  on  the 
diameter  that  bisects  the  chord,  and  its  distance  from  the 
centre  is  a  fourth  proportional  to  the  arc,  its  chord,  and 
the  radius. 

Centre  of  Gravity  of  a  Parabolic  Area. 

84.  Let  the  area  be  limited  by  a  double  ordinate,  and 
denote  the  extreme  abscissa  by  a.  From  the  equation  of 
the  curve,  y*  =  2px,  we  have, 


y  = 

/.  dv  =  ydx  =  V%p  •  x*dx, 

_  o 

and  xdv  —  v%p  -  ofidx. 
Substituting  in  (1).  and  integrating  between 
the  limits  0  and  a,  we  have, 

3 
*i  -  5  * 


APPLICATIONS   TO    MECHANICS   AND    ASTRONOMY.       1?5 

That  is,  the  centre  of  gravity  is  on  the  axis,  at  a  distance 
from  tlie  vertex  equal  to  three-fifths  the  altitude  of  Hie 
segment. 

Centre  of  Gravity  of  a  Semi-ellipsoid  of  Revolution. 

85.  Let  the  axis  of  the  ellipsoid  be  taken  as  the  axis 
of  x.  Then,  if  the  origin  be  taken  at  the  centre,  the 
equation  of  the  generating  curve  is 


In  this  case,  we  have, 

Bu 

dv  —  vy*dx  =  *  —  (a2  —  xz)dx, 

b2 

and  xdv  =  it  —-  (a2x  —  xs)dx. 
a2 

Substituting  in  (1),  and  integrating  between  the  limits 
x  —  0,  and  x  =  a,  we  have, 


That  is,  the  centre  of  gravity  of  a  semi-prolate  spheroid 
of  revolution  is  on  its  axis  of  revolution,  and  at  a  distance 
from  the  centre  equal  to  three-sixteenths  the  major  axis  of 
the  generating  ellipse. 

If  we  change  a  to  #,  and  b  to  a,  we  find  for  the  semi- 
oblate  spheroid, 

3.        3  _ 


176  DIFFERENTIAL   AND   INTEGRAL   CALCULUS.      . 

Centre  of  Gravity  of  a  Cone. 

86.  Let  the  axis  of  the  cone  be  the  axis  of  x,  and  the 
vertex  of  the  cone  at  the  origin ;  denote  the  altitude  by  h, 
and  the  radius  of  the  base  by  r  ;  then  will  the  tangent  of  the 

•• 

semi-angle  of  the  cone  be  •=-,  the  equation  of  the  generating 

li 

T 

line  will  be  y  —     x,  and  we  have, 


dv  =  *yzdx  =  if  7-r  x*dx,  and  xdv  = 

li*  if 

Substituting  in  (1),  and   integrating  between  the  limits 

0  and  h,  we  have, 

_  3 

Centre  of  Gravity  of  a  Paraboloid  of  Revolution. 

87.  Let  the  axis  of  the  paraboloid  be  taken  as  the  axis  of 
x.     The  equation  of  the  parabola  being  y2  =  2px,  we  have, 

;,  and  xdv  = 


Substituting  in  (1),  and  integrating  from  x  —  0  to  x  =  at 

we  have, 

2 


II.  MOMENT  OF  INERTIA. 

Definitions  and  Preliminary  Principles. 

88.  The  moment  of  inertia  of  a  body  with  respect  to  an 
axis,  is  equal  to  the  algebraic  sum  of  the  products  obtained 
by  multiplying  the  mass  of  each  element  of  the  body  by 
the  square  of  its  distance  from  the  axis.  If  we  take  the 
axis  through  the  centre  of  gravity  of  the  bodv.  and  denote 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      177 

the  mass  of  an  element  by  dm,  its  distance  from  the  axis 
by  z,  and  the  moment  of  inertia  by  IY,  we  have, 


K=J**dm (l) 

If  we  take  any  parallel  axis  at  a  distance  d  from  the 
assumed  axis,  and  denote  the  moment  of  inertia  with 
respect  to  it  by  K',  we  have  (Mechanics,  Art.  123). 

K'=  K  +  md* (2) 

Moment  of  Inertia  of  a  Straight  Line. 

89.  Let  the  axis  be  taken  through  the  centre  of  gravity 

of  the  line  and  perpendicular  to  it.  Let  AB  represent  the 
line,  CD  the  axis,  and  E  any  ele- 

ment.      Denote  the  length  of  the  |                 [ 

line  by  21,  its  mass  by  m,  the  dis-  ~~G jf 

tance    GE  by  x,  and   the   length  *""              JcT"1" 

of  the  element  by  dx.     From  the  Fis- 19- 
principle  of  homogeneity,  we  have, 

21  :  dx  :  :  m  :  dm ;   /.  dm  =  —  dx. 

Substituting  in  (1),  and  integrating  from  x  =  —  I  to 
x  =  +  I,  we  have, 

K  —m-\  .'.  K'  =  m  ( —  +  d* ) (3) 

<J  \O  / 

These  formulas  are  entirely  independent  of  the  breadth 
of  AB  in  the  direction  of  the  axis  CD ;  they  hold  good, 
therefore,  when  the  filament  is  replaced  by  the  rectangle 
EF9  the  axis  being  parallel  to  one  of  its  ends.  In  this 
case  m  is  the  mass  of  the  rectangle ;  21,  its  length ;  and  d, 
the  distance  of  the  axis  from  the  centre  of  gravity  of  the 
rectangle. 


178 


DIFFERENTIAL  AND   INTEGRAL   CALCULUS. 


Moment  of  Inertia  of  a  Circle. 

90.  First)  let  the  axis  be  taken  to  coincide  with  one  of 
the  diameters.  Let  ACB  represent  the  circle,  AB  the 
axis,  and  C'D'  an  element  parallel  to  A 

the  axis.  Denote  00  by  r,  OE  by  x, 
EF  by  dx,  C'D'  by  2y,  and  the  mass 
of  the  circle  by  m.  Then,  because  the 
circle  is  homogeneous,  we  have, 


•BT 


:  2ydx  :  :  m  :  dm  ; 


=  ^dx  = 

-TIT2 


Substituting  in  (1), 


f(r*  - 
*J 


Keducing  by  Formulas  A  and  B,  integrating  by  (20), 
and  taking  the  integral  between  the  limits  x  =  —  r  and 
r.  =  +  r,  we  find, 


(4) 


Secondly,  let  the  axis  be  taken  through  the  centre  and 
perpendicular  to  the  plane  of  the  circle.  Let  KL  be  an 
elementary  ring,  whose  radius  is  x,  and 
whose  breadth  is  dx ;  then  will  its  area 
be  Zifxdx,  and  from  the  principle,  that  the  / 
masses  are  proportional  to  the  volumes, 
we  have, 


m  :  dm  ;   .*.  dm  =  — — j 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY. 


Substituting  in  (1),  and  integrating  from  x  =  0  to  x  =  r, 
we  have, 


(5) 


Thirdly,  to  find   the  moment  of  inertia  of  a  circular 
ring   with  respect  to  an  axis  perpendicular  to  its  plane. 
Let  m  denote  the  mass  of  the  ring,  r  and  r ' 
its  extreme  radii ; 

Then,  7t(r2  —  r'2)  :  %nxdx  ::m:  dm  ; 

.\dm  =  ^ jj  dx. 

Substituting   in   (1)   and  integrating  from 
x  =  r'  to  x  =  r,  we  have 


Fig.  22. 


Formulas  (5)  and  (6)  are  independent  of  the  thickness 
of  the  plate ;  hence,  they  will  be  true  whatever  that  thick- 
ness may  be.  Hence,  they  hold  good  for  a  solid  and  hollow 
cylinder,  m  being  taken  to  represent  the  mass  of  the 
cylinder. 

Moment  of  Inertia  of  a  Cylinder  with  respect  to  an  Axis 
perpendicular  to  the  Axis  of  the  Cylinder. 

91.  Let  the  axis  be  taken  through  the  centre  of  gravity 
of  the  cylinder,  and  let  EF  be  an  element  perpendicular 
to  the  axis  of  the  cylinder.  Denote  the 
length  of  the  cylinder  by  21,  its  radius 
by  r,  its  mass  by  m,  the  distance  of  EF 
from  the  axis  by  x,  and  the  thickness 
by  dx.  Then,  as  before,  we  have, 


dx  : :  m  :  dm  ; 


7         m 
dm  =  r 


180 


DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 


The  moment  of  inertia  of  this  element  is  given  by  For- 
mula (4)  ;  but  this  is  the  differential  of  the  moment  of 
inertia  of  the  entire  cylinder;  hence,  substituting  dm  for 
m,  and  x  for  d,  in  (4),  we  have, 


Integrating  from  —  I  to  +  Z,  we  have, 


(7) 


Moment  of  Inertia  of  a  Sphere. 

92.  Let  the  axis  pass  through  the  centre  of  a  sphere. 
Let  C'D'  be  an  elementary  segment  perpendicular  to 
the  axis,  DC,  whose  volume  is  *y2dx 
=  *(r2  —  x2)dx;  its  mass  is  found  -- — r^-c' 

from  the  proportion, 


- 


:  if(r2  —  x2)dx  :  :  m  :  dm; 


.:  dm 


3m 


(r2  -x2)dx. 


Substituting  this  value  of  dm  for  m  in  equation  (5),  and 
making  r  in  that  equation  equal  to  y,  or  to  Vr2  —  x2,  we 
have,  for  the  differential  of  the  moment  of  inertia, 


^ 

=(r2  -x*)2dx. 


Integrating  between  the  limits  x  =  —  r,  and  x  =  +  r,  we 
have, 


APPLICATIONS  TO   MECHANICS  AND   ASTRONOMY.      181 

III.  MOTION  OF  A  MATERIAL  POINT. 

General  Formulas. 

93.  Uniform,  motion  is  that  in  which  the  moving  point 
passes  over  equal  spaces  in  equal  times  ;  variable  motion 
is  that  in  which  the  moving  point  passes  over  unequal 
spaces  in  equal  times.  The  velocity  of  a  point  is  its  rate 
of  motion.  The  acceleration  due  to  a  force  is  the  rate  of 
change  that  it  produces  in  the  velocity  of  a  point.  When 
a  force  acts  to  increase  the  velocity,  the  acceleration  is 
positive)  when  it  acts  to  diminish  the  velocity,  the  accelera- 
tion is  negative,  and  conversely. 

Let  us  denote  the  mass  of  a  material  point  by  m,  its 
velocity  at  any  time  t,  by  v,  the  space  moved  over  at  the 
time  t,  by  s,  and  the  acceleration  due  to  the  moving  force 
by  <p. 

When  the  motion  is  uniform,  the  velocity  is  constant,  and 
its  measure  is  the  space  passed  over  in  any  time  divided  by 
that  time  ;  but  we  may,  in  all  cases,  regard  the  motion  as 
uniform  for  an  infinitely  small  time  dt  ;  denoting  the 
space  described  in  that  time  by  ds9  we  have, 

ds 


When  the  velocity  varies  uniformly  the  acceleration  is 
constant,  and  we  take  for  its  measure  the  change  of 
velocity  in  any  time  divided  by  that  time;  but  we  may,  in 
all  cases,  regard  the  velocity  as  varying  uniformly  for  an 
infinitely  small  time  dt  ;  denoting  the  change  of  velocity 
in  that  time  by  dv,  we  have, 


182  DIFFERENTIAL  AND   INTEGRAL   CALCULUS. 

In  the  discussion  of  motion  it  is  customary  to  regard  the 
time  as  the  independent  variable.  Differentiating  (1), 
under  that  supposition,  we  find, 


substituting  this  in  (2),  we  have, 


Equations  (1),  (2),  and  (3)  are  fundamental,  and  from 
them  we  may  deduce  many  laws  of  motion.  If  we  multi- 
ply both  members  of  (3)  by  m,  we  have, 

cl2s  p  _     d2s 

In  (4),  F  is  the  moving  force.  It  will  often  be  conve- 
nient to  regard  the  mass  of  the  material  point  as  the  unit 
of  mass,  in  which  case,  we  have,  m  =  1. 

Uniformly  Varied  Motion. 

94.  Uniformly  varied  motion  is  that  in  which  the  ve- 
locity increases,  or  diminishes,  uniformly.  In  the  former 
case  the  motion  is  uniformly  accelerated,  in  the  latter  it  is 
uniformly  retarded,  in  both  the  acceleration  is  constant. 
Denoting  the  constant  value  of  <p  by  g,  substituting  in  (3), 
multiplying  both  meml  jrs  by  dt,  and  integrating,  we 
have, 

^  =  fff.  +  C;  or,  v  =  gt  f  C (5) 

Multiplying  again  by  dt,  and  integrating,  we  find, 
s  =  \gt*  +  Ct  +  C1 (6) 


APPLICATIONS   TO   MECHANICS   AND    ASTRONOMY.      183 

The  constant,  (7,  is  what  v  becomes  when  t  =  0 ;  this  is 
called  the  initial  velocity,  and  may  be  denoted  by  v'.  The 
constant,  0',  is  what  s  becomes  when  t  =  0 ;  this  is  called 
the  initial  space,  and  may  be  denoted  by  s'.  Making  these 
substitutions  in  (5)  and  (6),  we  have, 

v  =  v'+fft (7) 

*  =  s'  4-  v't  +  \gt* (8) 

Equations  (7)  and  (8)  enable  us  to  discuss  all  the  cir- 
cumstances of  uniformly  varied  motion  (see  Mechanics, 
Arts.  103  to  108).  If  g  represent  the  force  of  gravity,  and 
v'  and  s'  be  each  equal  to  0,  (7)  and  (8)  will  express  the 
laws  of  motion  when  a  body  falls  from  rest  under  the  in- 
fluence of  gravity,  regarded  as  a  constant  force.  Under 
this  supposition  they  become, 

*=fft (9) 

s  =  W (10) 

That  is,  the  velocities  generated  are  proportional  to  tne 
times,  and  the  spaces  fallen  through  are  proportional  to 
the  squares  of  the  times. 

Bodies  Falling  under  the  Influence  of  Gravity,  regarded  as 
Variable. 

95.  In  accordance  with  the  Newtonian  law,  the  attrac- 
tion exerted  by  the  earth  on  a  body  at  different  distances 
varies  inversely  as  the  squares  of  their  distances.  Denoting 
the  radius  of  the  earth,  supposed  a  sphere,  by  r,  the  force 
of  gravity  at  the  surface  by  g,  any  distance  from  the  cen- 
tre, greater  than  r,  by  s,  and  the  force  of  gravity  at  that 
distance  by  <p,  we  have, 

3        *  ~ 


184  DIFFERENTIAL   AND    INTEGRAL    CALCULUS. 

Substituting  this  value  of  9  in  (3),  and  at  the  same  time 
making  it  negative,  because  it  acts  in  the  direction  of  s 
negative,  we  have, 


Multiplying  by   2ds,  and  integrating  both  members,  we 
have, 

ds*       2rz  Zr* 


If  we  make  v  =  0,  when  s  =  li,  we  have, 


aid,  »'=2<rr'j-        .....  (13) 

Equation  (13)  gives  the  velocity  generated  whilst  the 
body  is  falling  from  the  height  li  to  the  height  s.  If  we 
make  h  —  co  ,  and  s  =  r,  (13)  becomes, 


v*  =  2gr;     .-.  v  —      fyr  .....  (14) 

In  this  equation  the  resistance  of  the  air  is  not  con- 
sidered. 

If  in  (14)  we  make  g  =  32.088  feet,  and  r  —  20923590 
feet,  their  equatorial  values,  we  find, 

v  —  3GG44  feet,  or  nearly  1  miles  per  second. 

Equation  (14)  enables  us  to  compute  the  velocity  acquired 
by  a  body  in  falling  from  an  infinite  distance  to  the  sun. 
If  we  make  g  =  890.16  feet,  and  r  =  430854.5  miles,  which 
are  their  values  corresponding  to  the  sun,  we  find, 

v  =  381  miles  per  second. 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      185 

If  we  make  li  equal  to  the  distance  of  Neptune,  and  s 
equal  to  the  sun's  radius  in  (13),  we  find  the  velocity  that 
a  body  would  acquire  in  falling  from  Neptune  to  the  sun, 
under  the  influence  of  the  sun's  attraction. 

To  find  the  time  required  for  a  body  to  fall  through  any 

ds 

space,  substitute  —  for  v  in  (13),  and  solve  the  results  with 
cct 

respect  to  dt;  this  gives, 

^  =  V^'7^=~/^'vlb-(15) 

The  negative  sign  is  taken  because  s  decreases  as  t  in- 
creases. Reducing  (15)  by  Formula  E,  and  integrating 
by  Formula  (20),  we  find, 


If  t  —  0  when  s  =  h,  we  have, 


This,  in  (16),  gives, 

I  ftversin-1  J  +  fr*]  .  .  .  (17) 
Making  s  =  r  in  (17),  we  find, 

IT,  [(/<<•  -  r-)*  -  I  Avorsin-1}:  +  ^]  .  .  .  (18) 


Which  gives  the  time  required  for  a  body  to  fall  from  a 
distance  h,  to  the  surface  of  the  sun. 


186  DIFFERENTIAL   AND    INTEGRAL   CALCULUS. 


Bodies  Palling  under  the  Influence  of  a  Force  that  vaiies  as 
the  Distance. 

96.  It  will  be  shown  hereafter,  that  if  an  opening  were 
made  along  one  of  the  diameters  of  the  earth,  and  a  body 
permitted  to  fall  through  it,  the  body  would  be  urged 
toward  the  centre  by  a  force  varying  as  the  distance  from 
the  centre,  provided  the  earth  were  homogeneous.  As- 
suming that  principle,  and  denoting  the  force  at  the  sur- 
face by  g,  the  radius  being  r,  and  the  force  at  the  distance, 
x,  from  the  centre  by  <p,  we  have, 

r  :  s  :  :  g  :  <p  ; 

Substituting  this  value  of  <p  in  (3),  and  at  the  same  time 
giving  it  the  minus  sign,  because  it  acts  in  the  direction 
of  s  negative,  we  have, 


Multiplying  by  2ds,  and  integrating, 

|!=_i»!  +  (7;oj;)V.  =  (7-?,.. 

dtz  r  r 

Making  v  —  0,  when  s  =  r,  we  have,  C  —  ^  r2  ;  hence, 


-=-,«).        -.(19) 
cit*      r 


If  s  =  0,  we  find  v  =  Vgr,  which  is  the  maximum  velocity. 
It  is  eqjal  to  the  entire  velocity  generated  by  a  body  fall- 
ing from  an  infinite  distance  to  the  surface  of  the  earth 

divided  by  A/2.  If  the  body  pass  the  centre,  s  becomes 
amative,  and  we  find  the  same  values  for  v  at  equal  dis- 


APPLICATIONS  TO   MECHANICS   AND   ASTRONOMY.      18? 

tances  from  the  centre,  whether  s  be  positive  or  negative. 
When  s  becomes  equal  to  —  r,  v  reduces  to  0,  and  the 
body  then  falls  toward  the  centre  again,  and  so  on  con- 
tinually. 

Let  us  suppose  A'C'  to  be  the  diameter  along  which  the 
body  oscillates,  and  at  the  time  the  body  starts  from  A' 
let  a  second  body  start  from  the  same 
point  and  move  around  the  semi-cir- 
cumference, A' MO',  with  a  constant 
velocity  equal  to  *fgr.    At  any  point, 
M,  let  this  velocity  be  resolved  into 
two  components,  MQ  and  MN,  the 
former  perpendicular  and  the  latter 
parallel  to  A'C'.     The  latter  component  will  be  equal  to 
y  multiplied  by  cosTMN,  or  its  equal,  cosB'MII' ;  but 


COB&MH'  =  ^TJ,;    denoting  B'H'  by  s,  whence  H'M 
we  have,  cosB'MH' 

=  — — ;  hence,  NN—  A/ ^-  .  (r2  —  s*)* :  but  this 

r  I/    r 

is  equal  to  the  velocity  of  the  oscillating  point  when  at  IT. 
Hence,  the  velocity  of  the  vibrating  point  is  everywhere 
equal  to  the  parallel  component  of  the  velocity  of  the 
revolving  point;  they  will,  therefore,  come  together  at  the 
points  A  and  6r/,  and  the  position  of  the  vibrating  point 
will  always  be  found  by  projecting  the  corresponding  posi- 
tion of  the  revolving  point  on  the  path  of  the  former.  To 
find  the  time  for  a  complete  vibration  from  A'  to  C",  solve 
(19)  with  respect  to  dt,  whence, 


9     Vr*- 


(20) 


188  DIFFERENTIAL   AND    INTEGRAL    CALCL'LUS. 

The  negative  sign  is  used  because  t  is  a  decreasing  func- 
tion of  s.  Integrating  (20)  between  the  limits  s  =  r  and 
s  =  —  r,  we  find. 


t"  = 


This,  we  shall  see  hereafter,  is  the  time  of  vibration  of  a 
simple  pendulum  whose  length  is  r. 

The  time  maybe  found  otherwise,  as  follows:  The  space 
passed  over  by  the  revolving  point  is  ir-ry  dividing  this  by 
the  velocity,  V]/r,  we  have, 


t—    T 


The  species  of  vibratory  motion  just  discussed  is  some- 
times called  harmonic,  being  the  same  as  that  of  a  point 
of  a  vibrating  chord  or  spring. 

Vibration  of  a  Particle  of  an  Elastic  Medium. 

97.  It  is  assumed  that  if  a  particle  of  an  elastic  medium 
be  slightly  disturbed  from  its  place  of  rest,  and  then  aban- 
doned, it  will  be  urged  back  by  a  force  that  varies  directly 
as  the  distance  of  the  particle  from  its  position  of  equilib- 
rium ;  on  reaching  this  position,  the  particle,  by  virtue  of 
its  inertia,  will  pass  to  the  other  side,  again  to  be  urged 
back,  and  so  on. 

Let  us  denote  the  displacement,  at  any  time  t,  by  s,  and 
the  acceleration  due  to  the  restoring  force  by  <p ;  then,  from 
the  law  of  force,  we  have,  <p  =  n*s,  in  which  n  is  constant 
for  the  same  medium,  under  the  same  circumstances  of 
density,  pressure,  etc.  Substituting  for  9  its  value  from 
Equation  (3),  and  prefixing  the  negative  sign,  because  it 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      189 

acts  in  a  direction  contrary  to  that  in  which  s  is  estimated, 
we  have, 


Multiplying  by  2ds,  and  integrating,  we  have, 

-j£  =  n***  +  C=-V*  .....  (23) 

The  Telocity  is  0  when  the  particle  is  at  the  greatest 
distance  from  the  position  of  equilibrium;  denoting  this 
value  of  s  by  a,  we  have, 

n*a*  +  (7=0;     .-.  C=  -n*a*, 
which,  in  (23),  gives, 

^  =  n*(a*  -  «*)  ;  or,  ndt  =  —J^=  .....  (24) 
dt*  Vfl2  -  s2 

Integrating  (24),  we  have. 

nt  +  G'  =  sin"1-  .....  (25) 

Taking  the  sines  of  both  members,  and  reducing,  we  have, 
s  =  asui(nt  +  C)  .....  (26) 

If  we  make  t  =  0,  when  s  =  0,  we  have  (7  =  0,  ana  Equa- 
tion (2G)  becomes, 

s  =  a  sm(nt)  .....  (27) 

If  (25)  be  taken  between  the  limits  —  a  and  +  a,  we  find 
for  the  time  of  a  single  vibration,  denoted  by  \T, 


190  DIFFERENTIAL   AND    INTEGRAL    CALCULUS. 

Substituting  this  in  (27),  we  have,  finally, 

n   —   n  O1'11  /  /  I  /OC\ 

8  —  asiu\^~tj (28) 

in  which  T  is  the  time  of  a  double  vibration. 
Solving  (24),  we  have, 

-1* (29) 

Substituting  for  s,  in  (29),  its  value  from  (28),  we  find, 

v  =  nA/ a*  —  a2siii2f~tj  =  naA/1  —  sin2f> 

Whence,  we  have, 

v  =  nacos(~t\ (30) 

This  equation  is  used  in  discussing  the  laws  of  light, 
and  in  many  other  cases. 

Curvilinear  Motion  of  a  Point. 

98.  A  point  cannot  move  in  a  curve  except  under  the 
action  of  an  incessant  force,  whose  direction  is  inclined  to 
the  direction  of  the  motion.  This  force  is  called  the  de- 
flecting force,  and  can  be  resolved  into  two  components, 
one  in  the  direction  of  the  motion,  and  the  other  at  right 
angles  to  it.  The  former  acts  simply  to  increase  or  di- 
minish the  velocity,  and  is  called  the  tangential  force  ;  the 
latter  acts  to  turn  the  point  from  its  rectilinear  direc- 
tion, and  being  directed  toward  the  centre  of  curvature 
is  called  the  centripetal  force.  The  resistance  offered  by 
the  point  to  the  centripetal  force,  in  consequence  of  its 


APPLICATIONS   TO    MECHANICS   AND   ASTRONOMY.      191 


inertia,  is   equal  and  directly  opposed  to  the  centripetal 
force,  and  is  called  the  centrifugal  force. 

To  find  expressions  for  the  tangential  and  centripetal 
forces,  let  the  acceleration  due  to  the  deflecting  force  in  the 
direction  of  the  axis  of  x,  at  any 
time  t}  be  denoted  by  JT,  and  the 
acceleration  in  the  direction  of 
the  axis  of  y  by  Y.  Let  these  be 
resolved  into  components  acting 
tangentially  and  normally.  The 
algebraic  sum  of  the  tangential 
components  is  the  tangential  accel- 
eration denoted  by  T,  and  the 
algebraic  sum  of  the  normal  components  is  the  centripetal 
acceleration  denoted  by  JV.  Assuming  the  notation  of  the 
figure,  we  have, 

T  =  JT 


N  —  JTsind  —  I"cos0. 
But  from  Articles  (93)  and  (5),  we  have, 


and, 


- 

dt* 


dt* 


dx  dy 

cosd  =  —r  ;     smd  =  -/-. 
ds  d« 


Substituting  in  the  preceding  equations,  and  reducing  oil 
the  supposition  that  t  is  the  independent  variable,  and  bv 
means  of  Formula  (3),  Art.  (38),  we  have, 


_  _  d~x    dx      d2y    dy 
~  ~di*  '  T*  +  W  '  ts 


(31) 


192  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

-_  _  d*xdy  —  d*ydx 
dt*  .  ds 

dsz    d*ydx  —  d*xdy  __       v* 
~di*'~     ~ds*~  ~  ~R 

But  the  centripetal  acceleration  is  equal  and  directly  op- 
posed to  the  centrifugal  acceleration.  Denoting  the  lattei 
by  f,  we  have, 

v 2 


IVom  (31)  we  see  that  the  tangential  force  is  independ 
ent  of  the  centripetal  force,  and  from  (32)  we  see  that  the 
acceleration  due  to  the  centrifugal  force  at  any  point  of  the 
trajectory,  is  equal  to  the  square  of  the  velocity  divided  by 
the  radius  of  curvature  at  that  point. 

Velocity  of  a  Point  rolling  down  a  Curve  in  a  Vertical  Plane. 

90.  Let  a  point  roll  down  a  curve,  situated  in  a  vertical 
plane,  under  the  influence  of  gravity  regarded  as  constant. 
Let  the  origin  of  co-ordinates  be  taken  at  the  starting 
point,  and  let  distances  downward  be  positive.  At  any 
point  of  the  curve,  whose  ordinate  is  y,  the  force  of  gravity 
being  denoted  by  g,  we  have,  for  the  tangential  component 

of  gravity,  #sin<3,  or,  g  -f ;  placing  this  equal  to  its  value, 
as 

equation  (31),  we  have, 

dij      dzs 


Integrating  and  reducing,  remembering  that  the  constant 
ife  0  under  the  particular  hypothesis,  we  have, 


2  is*  ;  or?  v  = 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      193 

The  second  member  of  (1)  is  the  velocity  due  to  the 
height  y.  Hence,  the  velocity  generated  by  a  body  rolling 
down  a  curve,  gravity  being  constant,  is  equal  to  that  gen- 
erated by  falling  freely  through  the  same  vertical  height. 

This  principle  is  true  so  long  as  g  is  constant.  But  g 
may  be  regarded  as  constant  from  element  to  element,  no 
matter  what  may  be  the  law  of  variation.  Hence,  if  a  body 
fall  toward  the  sun,  or  earth,  on  a  spiral  line,  the  ultimate 
velocity  will  be  the  same  as  though  it  had  fallen  on  a  right 
line  toward  the  centre  of  the  attracting  body.  The  direc- 
tion of  the  motion,  however,  is  not  the  same,  for  in  the 
former  case  it  is  tangential  to  the  trajectory  pursued,  and 
in  the  latter  case  it  is  normal  to  the  attracting  body. 

The  Simple  Pendulum. 

100.  A  simple  pendulum  is  a  material  point  suspended 
from  a  horizontal  axis,  by  a  line  without  weight,  and  free 
to  vibrate  about  that  axis. 

Let  ABC  be  the  arc  through  which  the  vibration  takes 
place,  and  denote  its  radius  by  I.     The  angle  CD  A  is  the 
amplitude  of  vibration  ;  half  this  angle, 
ADB,  denoted  by  a,  is  the  angle  of 
deviation;   and  I  is  tlie  length  of  tlie 
pendulum.     If  the  point  start  from 
rest,  at  A,  it  will,  on   reaching  any 
point,  //,  of  its  path,  have  a  velocity,  v, 
due  to  the  height  EK,  denoted  by  y. 
Hence, 

v  = 


If  we  denote  the  angle  HDB  by  6,  we  have  DK  '  =  7cos0; 
we  also  have  DE=lcosa  •  and  since  y  is  equal  to  DK—DE, 
we  have, 

y  =  J(cos0  -  COSa), 
0 


194  DIFFERENTIAL    AND   INTEGRAL   CALCULUS. 

which,    being    substituted    in    the    preceding    formula, 
gives, 


v  =  A2#/(cos4  —  cosa). 


Again,  denoting  the  angular  velocity  at  the  time  i  by  <p, 

id  rememberi 
the  pendulum, 


and  remembering  that  9  =  — ,  we  have,  for  the  velocity  of 

Cit 


Equating  the  two  values  of  v,  and  reducing,  we.  have, 


—  cosa 

Developing  cos$  and  cosa  by  McLaurin's  Formula,  we  have, 

^ -?•+£'-•** 

a3         «* 
~  2~  f  24  ~ 

If  we  suppose  the  amplitude  to  be  small,  we  may  neglect 
all  the  terms  after  the  second ;  doing  so,  and  substituting 
in  (31),  we  have, 


Integrating  between  the  limits  &  ==  —  a,  and  A  =  +  a}  we 
find, 

t  = 


which  is  the  formula  for  the  time  of  vibration  of  a 
pendulum. 


APPLICATIONS  TO   MECHANICS   AND   ASTRONOMY.      195 


Attraction  of  Homogeneous  Spheres. 

101.  The  Newtonian  Law  of  universal  gravitation  may 
be  expressed  as  follows,  viz. :  every  particle  of  matter 
attracts  every  other  particle,  with  a  force  that  varies 
directly  as  the  mass  of  the  attracting  particle,  and  inversely 
as  the  square  of  the  distance  between  the  particles.  To 
apply  this  law  to  the  case  of  homogeneous  spheres,  let  us 
first  consider  the  action  of  a  spherical  shell,  of  infinitesimal 
thickness,  on  a  material  point  within  it. 

Let  D  be  the  material  point,  APB  a  great  circle  through 
it,  and  let  the  diameter  ADB  be  taken  as  the  axis  of  x. 
If  the  circle  be  revolved 
about  AB,  its  circum- 
ference will  generate  a 
spherical  shell,  and  any 
element  of  the  circnmfer- 
ence,  as  P,  will  generate 
an  elementary  zone  whose 
altitude  is  dx.  For  any 
point  of  this  zone,  as  P, 
there,  is  another  point,  P', 

symmetrical  with  it,  and  the  resultant  action  of  these  points 
on  D  is  directed  along  AB,  and  is  equal  to  the  sum  of  the 
forces  into  the  cosine  of  the  angle  BDP.  Hence,  the  re- 
sultant attraction  of  the  entire  zone  is  directed  along  AB, 
and  is  equal  to  the  sum  of  the  attractions  of  all  its  particles 
into  the  cosine  of  the  angle  BDP.  To  find  an  expression 
for  this  resultant,  denote  CP  by  r,  CD  by  a,  DP  by  z, 
and  C7?  by  x.  Because  the  shell  is  infinitesimal  in  thick- 
ness and  homogeneous,  the  mass  of  the  zone,  is  to  the  mass 
of  the  shell,  as  dx,  the  altitude  of  the  zone,  is  to  2r,  the 
altitude  of  the  shell.  Hence,  if  the  mass  of  the  shcil  be 


*9G  DIFFERENTIAL    AXD    INTEGRAL   CALCULUS. 

denoted  by  m,  the  mass  of  the  zone  will  be  denoced  by 
-—  --.     If  the  force  exerted  by  the  unit  of  muss  at  the  unit 

of  distance  be  taken  as  the  unit  of  force,  the  attraction 
exerted  by  all  the  particles  of  the  zone  on  the  point  D  will 

be  equal  to  ~~  .  —  ,  and  the  resultant  action  on  D,  in  the 

direction  of  AB,  denoted  by  df,  will  be  given  by  the  equa- 
tion, 

0 


From  the  triangle,  PCD,  we  have, 
DP*  =  z*  =  r2  +  «2  -  2ax  =  r2  -  az  -  2a(x  -  «), 
and  from  the  triangle,  EDP,  we  have, 

cos  ED  P  =  ^-^.. 

z 

Substituting  in  (1),  remembering  that  dx  equals  d(x  —  rt), 
we  have, 

m        .    (x  -  a]  d(x  -  a)  . 

f=Wr'  ---     --  5  .....  (  ' 

[r«  _  a*  -  2a(x  -  «)]» 

Regarding  (x  —  a)  as  a  single  variable,  reducing  by  For- 
mula A,  and  integrating  by  Formula  (29),  we  find, 


_  az  _  2a(x  -  a] 

+ 


Taking  the  integral  from  x  —  —  r  to  x  .-=  +  r,  we  have, 


APPLICATIONS   TO   MECHANICS   AND   ASfKOXOMY.       197 

Hence,  the  effect  of  the  attraction  of  the  shell  on  any 
point  within  it  is  null.  If  a  sphere  be  described  about  0 
as  a  centre  with  a  radius  equal  to  #,  we  may  call  that  paifc 
which  lies  between  it  and  the  surface  of  the  given  sphere 
the  exterior  sJicU,  and  the  sphere  itself  may  be  called  the 
nucleus.  From  what  precedes,  we  infer  that  any  point 
within  a  homogeneous  sphere  is  acted  on  by  the  sphere 
precisely  as  though  the  exterior  shell  did  not  exist.  Hence, 
a  point  at  the  centre  of  a  sphere  is  not  affected  by  the 
attraction  of  the  sphere. 

If  the  point  D'  be  taken  without  the  shell,  we  have, 

jyp2  =  z*  =  r*  +  fl2  +  2ax  =  r2  -  an  +  2a(x  +  a), 
and  from  the  triangle,  ED'P,  we  have, 

cosED'P  =  ^f  . 
z 

Substituting  in  (1),  we  have, 


1   [r2  -  cr  +  2a(x  +  a)]"2' 
Reducing  and  integrating  as  before,  we  find, 

-  _  m  (  .    x  -f  a 

r (  a\rz  —  az  +  %a(x  +  a^* 


)T 

,•2  .. 


+  C. 


-a8  -f  2a 


Taking    the   integral   between   the  -limits   x  =  —  r   and 
£=  +  ?•,  we  have,  when  a  >  r 


198  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

But  m  is  the  mass  of  the  shell,  and  a  is  the  distance  of  D 
from  the  centre;  hence,  the  shell  attracts  a  particle  with- 
out it  as  though  its  entire  mass  were  concentrated  at  its 
centre. 

A  homogeneous  sphere  may  be  regarded  as  made  np  of 
spherical  shells;  hence,  a  sphere  attracts  any  point  without 
it,  as  though  the  mass  of  the  sphere  were  concentrated  at  its 
centre.  The  same  is  true  of  a  sphere  made  up  of  homoge- 
neous strata,  which  vary  in  density  in  passing  from  the 
surface  to  the  centre.  It  is  to  be  inferred  that  two  homo- 
geneous spheres,  or  two  spheres  made  up  of  homogeneous 
strata,  attract  each  other  as  though  both  were  concentrated 
at  their  centres. 

If  an  opening  were  made  from  the  surface  to  the  centre 
of  the  earth,  supposed  homogeneous,  and  a  body  were  to 
move  along  it  under  the  earth's  attraction,  it  would  every- 
where be  urged  on  by  a  force  varying  diretily  as  the  dis- 
tance of  the  body  from  the  centre.  For,  denote  the  dis- 
tance of  the  body  from  the  centre  at  any  instant  by  x. 
The  body  will  only  be  acted  on  by  the  nucleus  whose  radius 
is  x.  If  we  take  r  to  represent  the  radius  of  the  earth, 
and  remember  that  the  masses  are  proportional  to  these 
volumes,  we  have, 

4  4 

M  :  m  : :  ^vr3  :  ^*x3  :  :  r3  :  x3. 
o  o 

Denoting  the  force  of  attraction  at  the  surface  by  g,  and 
the  force  of  attraction  at  the  point  whose  distance  from 
the  centre  is  x  by/,  we  have,  from  the  Newtonian  law, 


Hence,  the  proposition  is  proved.     (See  Art.  96.) 


APPLICATIONS   TO    MECHANICS   AND   ASTRO N'O.MV.      199 


Orbital  Motion. 

102.  If  a  moving  point,  P9  be  continually  acted  on  by  a 
deflecting  force  directed  toward  a  fixed  centre,  it  will  de- 
scribe a  line  or  path,  called  an  orbit.  If  the  moving  point 
be  undisturbed  by  the  action  of  any  other  force,  the  orbit 
will  lie  in  a  plane  passing  through  the  fixed  centre  and  an 
element  of  the  curve.  Let 
this  plane  be  taken  as  the 
co-ordinate  plane,  let  the 
fixed  point  be  the  origin, 
and  let  the  orbit  be  repre- 
sented by  APB. 

Denote  the  acceleration 
due  to  the  deflecting  force 
at  any  time  t  by/,  its  in- 


Fig.  29. 


clination  to  the  axis  by  9,  and  its  components  in  the  direc- 
tions of  the  co-ordinate  axes  by  A"  and  Y. 
From  the  figure,  we  have, 


X  —  ~/cos?,  and  Y=  —/ship 


If  we  regard  /  as  the  independent  variable,  dx  and  dy  will 
both  be  variable,  and  from  Equation  (3),  Art  93,  we  have, 


If  we  denote  the  co-ordinates  of  P  by  x  and  y,  and  ita 
radius  vector  FP  by  r,  we  have,  from  the  figure, 


x    .          ii 
cosp  =  -,  sm<p  =  -; 

T  T 


and 


+  y*  —  r2 ;    .-.  xdx  +  ydy  =  rdr. 


200  DIFFERENTIAL   AND    INTEGRAL   CALCULUS* 

Substituting  in  (1),  we  have, 


Multiplying  the  first  of  Equations  (2)  by  y,  the  second  by 
x}  and  subtracting  the  former  from  the  latter,  we  have, 

xd*y      ycPx  d  (xdy  —  ydx)       A 

w-V=°'  or        M    J=o-----(3) 

Multiplying  the  first  by  dx,  the  second  by  dy,  adding  the 
resulting  equations  and  reducing,  we  have, 


dxd*x      dydzy  _        4.xdx  +  ydy  _        fl 
~W       ~^dtT       ~J          F"  ~J 

Integrating  (3)  and  (4),  Art.  77,  we  have, 


and 


1  /7r2  4-  (In*  {* 

I^TP* +:*--/>*' 


or, 

^  +  2  ffdr  =  C" (6) 

Equations  (5)  and  (G)  make  known  the  circumstances 
of  motion  when  the  value  of/  is  given.  It  is  found  con- 
venient to  transform  them  to  a  system  of  polar  co-o Mi  nates, 
whose  pole  is  F,  and  whose  initial  line  is  FX.  The  for* 
mulas  for  transformation  are 

x  —  rcoep,  and  y  =  rsin<p (7) 

Hence,  by  differentiating,  we  have, 
du      dr  .  dq> 


and 

dx       dr                  .     dq>  ,n* 

=      cosf-win^ (9) 


APPLICATIONS  TO   MECHANICS   AND   ASTRONOMY.      201 

\V*e  also,  have,  Art.  (53), 

<&*  =  r*dq>*  +  dr*  ......  (10) 

Substituting  in  (5),  and  reducing  by  the  relations, 

%2    _j_  2/2 

aasiiKp  —  3,  cos?  =  0,  and  xcos?  +  f/sincp  =  ---  —  —  f, 
we  have, 


From  equation  (6),  we  have, 


But,  from  (11),  we  have, 

,,       rzdq> 
dt  =  ~C^ 

and  this,  in  (12),  gives, 


Equations  (11)  and  (13)  are  the  equations  required. 
Multiplying  (11)  by  dt,  and  integrating,  we  have, 


=  Ct  +  C'" (14) 


But,  by  Art.  53,  rzdy  is  twice  the  elementary  area  swept 
over  by  the  radius  vector;  hence,  the  first  member  of  (14) 
is  an  expression  for  twice  the  area  swept  over  by  the  radiua 
VHctor  up  to  the  time  t.  If  we  suppose  the  area  to  bo 
reckoned  from  the  initial  line  FX,  and  at  the  same  time 

9* 


202  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

suppose  t  =  0,  we  have  C'"  =  0.  Substituting  this  in  (14), 
and  in  the  resulting  equation  making  t  =  I,  we  find 
C  equal  to  twice  the  area  swept  over  in  the  first  unit  of 
time ;  denoting  the  area  described  in  the  unit  of  time  bv 
A,  we  have,  C  —  2 A,  and  consequently, 


=  2At (15) 


From  Equation  (15)  we  infer  that  the  areas  described  by 
elae  radius  vector  are  proportional  to  the  times  of  descrip- 
tion, and  this  without  reference  to  the  nature  of  the  de- 
flecting force. 

To  find  the  equation  of  the  orbit,  let  us  assume  as  M 
particular  case,  the  Newtonian  law  of  universal  gravita- 
tion. .Denoting  the  force  exerted  by  the  central  body  at  a 
unit's  distance  by  k,  we  have,  for  the  attraction  at  the 
distance  r, 


Making  r  =  -,  whence  dr  =  --  -,  and  substituting  for  (j 
*nd/  their  values  in  (13),  we  have, 


or, 

du2  2k 

_+w2=__ 


(17) 


V 


To  find  the  value  of  <7V,  let  ~  =  0,  when  r  =  r',  or 

op 

u  =  ur.     But  when  —  -  =  0,  we  also  have  -T—  =  0,  that  is. 


APPLICATIONS   TO   MECHANICS   AND   ASTRONOMY.      203 

the  radius  vector  is  perpendicular  to  the  arc.    If  we  denote 
the  corresponding  velocity  by  v',  we  have, 

and,  consequently, 

2k  2k 

±A*U    ~  v'*r'3' 
Making  these  substitutions  in  (18),  we  find, 

fiV  _  %k 

0        —  775   ~~  _./-/«• 


Denoting  this  value  of  Oy  by  S,  and  solving  Equation  (18), 
we  have, 

du*  2k     \  k* 


du*  (  2k     \ 

W  =  8'  -QI*)  = 


Solving  with  reference  to  <p,  and  taking  the  negative  sign 
of  the  radical,  we  have, 


VP2  —  (u  —  q)z  vpz  —  (u  —  q)2 

Integrating,  we  have, 

9  -  a  =  cos-1-^^  .....  (20) 

In  which  —  a  is  an  arbitrary  constant.  If  we  suppose 
9  —  0,  when  v  =  v\  r  =  rr,  and  u  =  u',  the  corresponding 
value  of  a  is  the  angular  distance  from  the  initial  line  to 
the  radius  vector  whose  direction  is  perpendicular  to  the 
element  of  the  curve. 


204:  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

Taking  the  cosines  of  both  members  of  (20),  we  have, 

— i  =  cos(<p  —  a),  or,  u  =  q  +  jocos(<p  —  a) 
and  finally  replacing  u  by  its  value,  and  solving,  we  have, 

r  = 


q  +  jt?cos(<p  —  a)      k  +      l6SA*  +  k2  X  cos(9  —  a) 


Equation  (21)  is  the  polar  equation  of  a  conic  section. 
Hence,  the  orbit  of  a  particle  about  a  central  attracting 
body  is  one  of  the  conic  sections.  If  the  orbit  is  a  closed 
curve,  it  must  be  an  ellipse.  This  corresponds  to  the  case 
of  a  planet  revolving  about  the  sun,  or  to  that  of  a  satellite 
revolving  about  its  primary. 

Law  of  Force. 

103.  If  the  orbit  of  a  revolving  particle  be  an  ellipse,  it 
must  be  attracted  to  the  focal  point  by  a  force  that 
varies  inversely  as  the  square  of  the  distance. 

The  polar  equation  of  an  ellipse  is 


. 

1  4-  ecos(<p  —  a) 

in  which  a  is  the  semi-  transverse  axis,  e  is  the  eccentricity, 
a  is  the  angular  distance  from  the  fixed  line  to  the  radius 
vector  drawn  to  the  nearest  vertex,  or  the  longitude  of  the 
perihelion  in  the  case  of  a  planetary  orbit.  The  angle, 
p  —  a,  is  the  angle  that  is  called  in  astronomy  the  true 
anomaly. 
Putting  r  =  -  ,  in  (1),  we  get, 


APPLICATIONS   TO    MECHANICS   AND   ASTRONOMY.      205 

Differentiating  twice,  with  respect  to  <p,  we  find, 

du  _  esin(cp—  a)                   . 

cfy  ~  "  ~a(l  -  ez) 
and, 

dzu  __  e  cos  (9  —  a)                   . 

~  2 


Adding  (2)  and  (4),  we  have, 
d*u  x 


/"\ 


Resuming  equation  (13)  of  the  last  article,  replacing  r  by 
its  value,  -,  and  dr 

'1C 

reducing,  we  have, 


its  value,  -,  and  dr  by  its  value,  --  r,  differentiating    and 

'1C  U 


Combining  (5)  and  (G),  and  solving  with  respect  to  /,  we 
Iiave, 

C*u*  C*          1_ 

/  -  a(l  -  e*)'  °r/  ~  a(l  -  e*)  '  r*  ' 

Hence,  f  varies  inversely  as  the  square  of  r,  which  was  to 
be  shown. 


Note  on  the  Methods  of  the  Calculus. 

104.  All  the  rules  and  principles  of  the  Calculus,  in  the. 
present  treatise,  have  been  deduced  in  accordance  with  the 
method  of  infinitesimals,  as  explained  in  Articles  6  and  7. 
They  might  also  have  been  deduced  by  the  method  of  limits. 
It  remains  to  be  shown  that  the  results  obtained  by  these 
two  methods  are  always  identically  the  same. 


206  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

To  explain  the  method  of  limits,  let  us  denote  any  func- 
tion of  x  by  y  ;  that  is,  let  us  assume  the  equation, 

y  =/(•*) (i) 

If  we  increase  x  by  a  variable  increment  h,  and  denote  the 
corresponding  value  of?/  by  y',  we  have, 

y'=f(x  +  h) (2) 

It  is  shown  in  Courteuay's  Calculus,  Article  4,  that  so  long 
as  x  retains  its  general  value,  the  new  state  of  the  function 
can  be  expressed  by  the  formula, 

y'  =f(x  +  U)  =f(x)  +  Ah  +  Bh*  +  Ch*  +  etc.  . .  (3) 

in  which  A,  B,  C,  etc.,  depend  on  x,  but  are  independent 
of  h.  If  we  subtract  (1)  from  (3),  member  from  member, 
we  have, 

y'  -y  =  Ah  +  3h*  +  (etc.)  h3 (4) 

Dividing  both  members  of  (4)  by  h,  we  have, 

^-^—  =  A  +  Bh  +  (etc.)  h* (5) 

The  first  member  of  (5)  is  a  symbol  to  express  the  ratio 
of  the  increment  of  the  variable)  to  the  corresponding  incre- 
ment of  the  function}  and  the  second  member  is  the  value 
of  that  ratio. 

It  is  shown  in  Algebra,  that  in  an  expression  like  the 
second  member  of  equation  (5),  it  is  always  possible  to 
give  to  h  a  value  small  enough  to  make  the  first  term 
numerically  greater  than  the  algebraic  sum  of  all  the 
others.  If  we  assign  such  a  value  to  h)  and  then  suppose 
h  to  go  on  diminishing,  the  second  member  will  contin- 
ually approach  A,  and  when  h  becomes  0,  the  second  mem- 


APPLICATIONS  TO   MECHANICS  AND   ASTRONOMY.      20? 

ber  will  reduce  to  A.     Hence,  A  is  a  quantity  towan1. 

i/'  —  ii 
which  the  ratio,  — j— ,  approaches  as  li  is  diminished,  but 

beyond  which  it  cannot  pass;  it  is  therefore  the  limit  of 
that  ratio.  This  limit  is  the  differential  coefficient  of  y, 
and,  as  may  easily  be  seen,  it  is  entirely  independent  of  dx. 
The  product  of  this  by  the  differential  of  the  variable  is 
Hie  differential  of  y. 

We  may,  therefore,  enunciate  the  method  of  limits  as 
follows:  viz.,  Give  to  the  independent  variable  a  variable 
increment,  and  find  the  corresponding  state  of  the  function  ; 
from  this  subtract  the  primitive  state,  and  divide  the  dif- 
ference by  the  variable  increment;  then  pass  to  the  limit 
of  the  quotient  by  making  the  increment  of  the  variable 
equal  to  0;  the  result  is  the  differential  coefficient  of  the 
function;  if  this  be  multiplied  by  the  differential  of  the 
variable,  the  product  is  the  differential  of  the  function. 

In  the  case  assumed,  we  have,  by  this  method, 

J£  =  A;  .:dy=Adx (6) 

If  we  make  h  —  dx,  in  Equation  (4),  dx  being  infinitely 
small,  the  first  member  will  be  the.  differential  of  y,  and 
all  the  terms  of  the  second  member  after  the  first  may  be 
neglected.  Hence,  by  the  method  of  infinitesimals,  we 
have, 

dy  =  Adx;    .:  -^  =  A. 
dx 

This  result  is  the  same  as  that  obtained  by  the  method  of 
limits.  But,  by  hypothesis,  y  represents  any  function  of 
x;  hence,  in  all  cases,  the  differential  coefficient  is  iden- 
tically the  same  whether  found  by  the  method  of  limits  or 
by  the  method  of  infinitesimals. 


208  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

Erery  function,  regarded  as  a  primitive,  is  connected 
with  some  other  function,  regarded  as  a  derivative,  by  the 
law  of  differentiation.  This  derivative  is  the  differential 
coefficient  of  the  primitive.  The  object  of  the  differential 
calculus  is  to  find  the  derivative  from  its  primitive;  the 
object  of  the  integral  calculus  is  to  find  the  primitive  from 
its  derivative;  every  application  of  the  calculus  depends 
on  one  of  these  processes,  or  on  some  discussion  growing 
out  of  one,  or  the  other.  Now,  because  the  primitive  and 
the  derivative  are  independent  of  the  differentials  of  both 
function  and  variable,4  the  relation  between  them  is,  of 
necessity,  independent  of  the  methods  employed  in  estab- 
lishing that  relation.  But  it  has  been  shown  that  the 
same  relation  is  found  between  these  functions,  whether 
we  employ  the  method  of  infinitesimals  or  the  method  of 
limits.  Hence,  these  methods,  and  the  results  obtained  bj 
them,  are  in  all  cases  logically  identical. 


BARNES'S     POPULAR     HISTORY 
OF    THE    UNITED    STATES. 


By  the  author  of  Barnes's  "Brief  Histories  for  Schools."  Complete  in  one  superb 
royal  octavo  volume  of  800  pages.  Illustrated  with  320  wood  engravings  and  14  steel 
plates,  covering  the  period  from  the  Discovery  of  America  to  the  Accession  of  President 
Arthur. 

Part  I.  Colonial  Settlement ;  Exploration  ;  Conflict ;  Manners  ;  Customs  ;  Educa- 
tion ;  Religion,  &c.,  &c.,  until  political  differences  with  Great  Britain  threatened  open 
rupture. 

Part  II.  Resistance  to  the  Acts  of  Parliament ;  Resentment  of  British  Policy,  and 
the  Succeeding  War  for  American  Independence. 

Part  III.  From  the  Election  of  President  Washington  to  that  of  President  Lincoln, 
with  the  expansion  and  growth  of  the  Republic  ;  its  Domestic  Issues  and  its  Foreign 
Policy. 

Part  IV.      The  Civil  War  and  the  End  of  Slavery. 

Part  V.  The  New  Era  of  the  Restored  Union  ;  with  Measures  of  Reconstruction  ; 
the  Decade  of  Centennial  Jubilation,  and  the  Accession  of  President  Arthur  to  Office. 

Appendix.  Declaration  of  Independence ;  The  Constitution  of  the  United  States 
and  its  Amendments ;  Chronological  Table  and  Index  ;  Illustrated  History  of  the 
Centennial  Exhibition  at  Philadelphia. 

The  wood  and  steel  engravings  have  been  expressly  chosen  to  illustrate  the  customs  of 
the  periods  reviewed  in  the  text.  Ancient  houses  of  historic  note,  and  many  portraits  of 
early  colonists,  are  thus  preserved,  while  the  elaborate  plans  of  the  Exposition  of  1876 
are  fully  given.  The  political  characteristics  of  great  leaders  and  great  parties,  which 
had  been  shaped  very  largely  by  the  issues  which  belonged  to  slavery  and  slave  labor, 
have  been  dealt  with  in  so  candid  and  impartial  a  manner  as  to  meet  the  approval  of 
all  sections  of  the  American  people.  The  progress  of  science,  invention,  literature,  and 
art  is  carefully  noted,  as  well  as  that  of  the  national  physical  growth,  thus  condensing 
into  one  volume  material  which  is  distributed  through  several  volumes  in  larger  works. 
Outline  maps  give  the  successive  stages  of  national  expansion ,  and  special  attention 
has  been  given  to  those  battles,  by  land  and  sea,  which  have  marked  the  military  growth 
of  the  republic.  JE1P"  Specially  valuable  for  reference  in  schools  and  households. 


From  Prof.  F.  F.  BARROWS,  Brown  School, 

Hartford,  Conn. 

"  Barnes's  Popular  History  has  been  in 
onr  reference  library  for  two  years.  Its 
concise  and  interesting  presentation  of 
historical  facts  causes  it  to  be  so  eagerly 
read  by  our  pupils,  that  we  are  obliged  to 
duplicate  it  to  supply  the  demand  for  its 
use." 

From  Hon.  JOHN  R.  BUCK. 
41 1  concur  in  the  above." 

From  Hon.  J.    C.    STOCK  WELL. 
"I  heartily  concur  with  Mr.  Barrows  in 
the  within  commendation    of    '  Barnes's 
Popular  History,'  as  a  very  interesting  and 
instructiA'e  book  of  reference." 
From  A.  MORSE,  Esq. 
"  I  cordially  concur  in  the  above." 

From  Rev.  WM.  T.  GAGE. 
"I  heartily    agree  with    the   opinions 
above  expressed." 

From  DAVID  CRARY,  Jr. 
"The  best  work  for  the  purpose  pub- 
lished." 


From  Prof.  S.  T.  DUTTON,  Superintendent 
of  Schools,  New  Haven,  Conn. 

"  It  seems  to  me  to  be  one  of  the  best 
and  most  attractive  works  of  the  kind  I 
have  ever  seen,  and  it  will  be  a  decided 
addition  to  the  little  libraries  which  we 
have  already  started  in  our  larger 
schools." 

From  Prof.  WM.  MARTIN,  of  Beattystown. 

N.J. 

"This  volume  is  well  adapted  to  the 
wants  of  the  teacher.  A  concise,  well- 
arranged  summary  of  events,  and  just  the 
supplement  needed  by  every  educator  who 
teaches  American  history." 

From  Prof.  C.  T.  R.  SMITH,  Principal  of 
the  Lansingburgh,  N.  F.,  Academy. 

"  In  the  spring  I  procured  a  copy  of 
'  Barnes's  Popular  History  of  the  United 
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SCHOOL  AND  COLLEGE  TEXT-BOOKS. 
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latter  gives  place  to  an  entirely  new  method  of  progressive  development,  based 
upon  some  of  the  best  features  of  the  word  system,  phonetics,  and  object 


THE  NATIONAL   SERIES   OF  STANDARD  SCHOOL-BOOKS, 

PARKER  &  WATSON'S  NATIONAL 
READERS. 


The  salient  features  of  these  works  which  have  combined  to  render  them  so  popular 
may  be  briefly  recapitulated,  as  follows  :  — 

x.  THE  WORD-BUILDING  SYSTEM.  — This  famous  progressive  method 
for  young  children  originated  and  was  copyrighted  with  these  books.  It  constitutes  a 
process  with  which  the  beginner  with  words  of  one  letter  is  gradually  introduced  to 
additional  lists  formed  by  prefixing  or  affixing  single  letters,  and  is  thus  led  almost 
insensibly  to  the  mastery  of  the  more  difficult  constructions.  This  is  one  of  the  most 
striking  modern  improvements  in  methods  of  teaching. 

2.  TREATMENT    OF    PRONUNCIATION.  — The    wants    of  the    youngest 
scholars  in  this  department  are  not  overlooked.     It  may  be  said  that  from  the  first 
lesson  the  student  by  this  method  need  never  be  at  a  loss  for  a  prompt  and  accurate 
rendering  of  every  word  encountered. 

3.  ARTICULATION  AND  ORTHOEPY  are  considered  of  primary  importance. 

4.  PUNCTUATION  is  inculcated  by  a  series  of  interesting  reading  lessons,  the 
simple  perusal  of  which  suffices  to  fix  its  principles  indelibly  upon  the  mind. 

5.  ELOCUTION.  —  Each  of  the  higher  Readers  (3d,  4th,  and  5th)  contains  elabo- 
rate, scholarly,  and  thoroughly  practical  treatises  on  elocution.     This  feature  alone  lias 
secured  for  the  series  many  of  its  warmest  friends. 

6.  THE   SELECTIONS  are  the  crowning  glory  of  the  series.     Without  excep- 
tion it  may  be  said  that  no  volumes  of  the  same  size  and  character  contain  a  collection 
so  diversified,  judicious,  and  artistic  as  this.     It  embraces  the  choicest  gems  of  Eng- 
lish literature,  so  arranged  as  to  afford  the  reader  ample  exercise  in  every  department 
of  style.     So  acceptable  has  the  taste  of  the  authors   in  this  department  proved,  not 
only  to  the  educational  public  but  to  the  reading  community  at  large,  that  thousands 
of  copies  of  the  Fourth  and  Fifth  Readers  have  found  their  way  into  public  and  private 
libraries  throughout  the  country,  where  they  are  in  constant  use  as  manuals  of  litera- 
ture, for  reference  as  well  as  perusal. 

7.  ARRANGEMENT.  —  The  exercises  are  so  arranged  as  to  present  constantly 
alternating  practice  in  the  different  styles  of  composition,  while  observing  a  definite 
plan   of  progression   or  gradation   throughout  the  whole.     In  the  higher  books  the 
articles  are  placed  in  formal  sections  and  classified  topically,  thus  concentrating  the 
interest  and  inculcating  a  principle  of  association  likely  to  prove  valuable  in  subse- 
quent general  reading. 

8.  NOTES   AND  BIOGRAPHICAL  SKETCHES.  -  These  are  full  and  ade- 
quate to  every  want.     The  biographical  sketches  present  in  pleasing  style  the  history  of 
every  author  laid  under  contribution. 

9.  ILLUSTRATIONS.  —These  are  plentiful,  almost  profuse,  and  of  the  highest 
character  or'  art.     They  are  found  in  every  volume  of  the  series  as  far  as  and  including 
the  Third  Reader. 

ip.  THE  GRADATION  is  perfect.  Each  volume  overlaps  its  companion  pre- 
ceding or  Mlowing  in  the  series,  so  that  the  scholar,  in  passing  from  one  to  another,  is 
only  conscious,  by  the  presence  oi'  the  new  book,  of  the  transition. 

11.  THE   PRICE  is  reasonable.     The  National  Readers  contain  more  matter  than 
any  other  series  in  the  same  number  of  volumes  published.     Considering  their  com- 
pleteness and  thoroughness,  they  are  much  the  cheapest  in  the  market. 

12.  BINDING.  — By  the  use  of  a  material  and  process  known  only  to  themselves, 
in  common  with  all  the  publications  of  this  house,  the  National  Readers  are  warranted 
to  outlast  any  with  which   they  may  be  compared,  the  ratio  of  relative  durability 
being  in  their  favor  as  two  to  one. 


NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 


SUPPLEMENTARY     READING. 


Monteith's    Popular    Science    Reader. 

James  Monteith,  author  of  Monteith's  Geographies,  has  here  presented  a  Supple- 
mentary Reading  Book  expressly  for  the  work  of  instruction  in  leading  and  science  at 
on«  and  the  same  time.  It  presents  a  number  of  easy  and  interesting  lessons  on  Natural 
Science  and  Natural  History,  interspersed  with  appropriate  selections  in  prose  and 
poetry  from  standard  authors,  with  blackboard  drawing  and  written  exercises.  It 
serves  to  instil  the  noblest  qualities  of  soul  and  mind,  without  rehearsing  stories  of 
moral  and  mental  depravity,  as  is  too  often  done  in  juvenile  books.  The  book  is  elabo- 
rately illustrated  with  fine  engravings,  and  brief  notes  at  the  foot  of  each  page  add  to 
the  value  and  teachableness  of  the  volume.  12mo,  half  bound,  360  pages. 

The    Standard    Supplementary    Readers. 

The  Standard  Supplementary  Readers  (formerly  Swinton's  Supplementary  Readtrs), 
edited  by  William  Swinton  and  George  R.  Cathcart,  have  been  received  with  marked 
favor  in  representative  quarters  from  Maine  to  California.  They  comprise  a  series  of 
carefully  graduated  reading  books,  designed  to  connect  with  any  series  of  school  Readers. 
They  are  attractive  in  appearance,  are  bound  in  cloth,  and  the  first  four  books  are 
profusely  illustrated  by  Fredericks,  White,  Dielman,  Church,  and  others.  The  six  books, 
which  are  closely  co-ordinated  with  the  several  Readers  of  any  regular  eeries,  are  •  — 

1.  Easy  Steps  for  Little  Feet.    Supplementary  to  First  Reader. 

In  this  book  the  attractive  is  the  chief  aim,  and  the  pieces  have  been  written  and 
chosen  with  special  reference  to  the  feelings  and  fancies  of  early  childhood.  128  pages, 
bound  in  cloth  and  profusely  illustrated. 

2.  Golden    Book    of    Choice    Reading.     Supplementary   to   Second 

Reader. 

This  book  represents  a  «;reat  variety  of  pleasing  and  instructive  reading,  consisting  of 
child-lore  and  poetry,  nobU*  examples  and  attractive  object-reading,  written  specially  for  it. 
1U2  pages,  cloth,  with  numerous  illustrations 

3      Book  of  Tales.    Being   School  Readings   Imaginative  and  Emotional. 

Supplementary  to  Third  Reader. 

In  this  book  the  youthful  taste  for  imaginative  and  emotional  is  fed  with  pure  and  noble 
creations  drawn  from  the  literature  of  all  nations.  272  pages,  cloth.  Fully  illustrated. 

4.  Readings  in  Nature's  Book.    Supplementary  to  Fourth  Reader. 
This  book  contains  a  varied  collection  of  charming  readings  in  natural  history   and 

botany,  drawn  from  the  works  of  the  great  modern  naturalists  and  travellers.    352  pages,, 
tloth.     Fully  illustrated. 

5.  Seven  American  Classics. 

6.  Seven  British  Classics. 

The  "  Classics  "  are  suitable  for  reading  in  advanced  grades,  and  aim  to  instil  a 
taste  for  the  higher  literature,  by  the  presentation  of  gems  of  British  and  American 
authorship.  220  pages  each,  cloth. 

8 


THE  NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


ORTHOGRAPHY. 

Smith's  Series. 

Smith's  Series  supplies  a  Speller  for  every  class  in  graded  schools,  and  comprises 
the  most  complete  and  excellent  treatise  on  English  Orthography  and  its  companion 
branches  extant. 

1.  Smith's  Little  Speller. 

First  round  in  the  ladder  of  learning. 

2.  Smith's  Juvenile  Definer. 

Lessons  composed  of  familiar  words  grouped  with  reference  to  similar  significa- 
tion or  use,  and  correctly  spelled,  accented,  and  defined. 

3.  Smith's  Grammar-School  Speller. 

Familiar  words,  grouped  with  reference  to  the  sameness  of  sound  of  syllables  dif- 
ferently spelled.  Also  definitions,  complete  rules  for  spelling  and  formation  of  deriva- 
tives, and  exercises  in  false  orthography. 

4.  Smith's  Speller  and  Definer's  Manual. 

A  complete  School  Dictionary,  containing  14,000  wor**-,  with  various  othei'  useful 
matter  in  the  way  of  rules  and  exercises. 

5.  Smith's  Etymology  —  Small  and  Complete  Editions. 

The  first  and  only  Etymology  to  recognize  the  Anglo-Saxon  our  mother  tongue; 
containing  also  full  lists  of  derivatives  from  the  Latin,  Greek,  Gaelic,  Swedish,  Norman, 
&c. ,  &c.  ;  being,  in  fact,  a  complete  etymology  of  the  language  for  schools. 

Northend's  Dictation  Exercises. 

Embracing  valuable  information  on  a- thousand  topics,  communicated  in  snch  a 
manner  as  at  once  to  relieve  the  exercise  of  spelling  of  its  usual  tedium,  and  combine 
it  with  instruction  of  a  general  character  calculated  to  profit  and  amuse. 

Phillip's  Independent  Writing  Spellers 

1.    Primary.  2.    Intermediate.  3.    Advanced. 

Unquestionably  the  best  results  can  be  attained  in  writing  spelling  exercises.  This 
series  combines  with  written  exercise  a  thorough  and  practical  instruction  in  penman- 
ship. Copies  in  capitals  and  small  letters  are  set  on  every  page.  Spaces  for  twenty 
words  and  definitions  and  errors  are  given  in  each  lesson.  In  the  advanced  book  there 
is  additional  space  for  sentences.  In  practical  life  we  spell  only  when  we  write. 

Brown's  Pencil  Tablet  for  Written  Spelling. 

The  cheapest  i 
64  lessons  of  25  w 

Pooler's  Test  Speller. 

The  best  collection  of  "  hard  words  "  yet  made.  The  more  uncommon  ones  are  fully 
defined,  and  the  whole  are  arrant/ed  alphabetically  for  convenient  reference.  The  book 
is  designed  for  Teachers'  Institutes  and  "  Spelling  Schools,"  and  is  prepared  by  an 
experienced  and  well-known  conductor  of  Institutes. 

Wright's  Analytical   Orthography. 

This  standard  work  is  popular,  because  it  teaches  the  elementary  sounds  ir  a 
plain  and  philosophical  manner,  and  presents  orthography  and  orthoepy  in  an  easy. 
uniform  system  of  analysis  or  parsing. 


The  cheapest  prepared  pad  of   ruled  blanks,  with  stiff  board  back,  sufficient  for 
ords. 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


ORTHOGRAPHY  —  Continued. 

Barber's  Complete  Writing  Speller. 

"The  Student's  Own  Hand-Book  of  Orthography,  Definitions,  ana  Sentences,  con- 
sisting  of  \\ritten  Exercises  in  the  Proper  Spelling,  Meaning,  and  Use  of  Words." 
(Published  1873.)  This  differs  from  Sherwood's  and  other  writing  spellers  in  its  more 
comprehensive  character.  Its  blanks  arc.  adapted  to  writing  whole  sentences  instead 
of  detached  words,  with  the  proper  divisions  for  numbering  corrections,  &c.  Such 
aids  as  this,  like  Watson's  Child's  Speller  and  Phillip's  Writing  Speller,  find  their 
raison  d'etre  in  the  postulate  that  the  art  of  correct  spelling  is  dependent  upon  written, 
and  not  upon  spoken  language,  for  its  utility,  if  not  for  its  very  existence.  Hence 
the  indirectness  of  purely  oral  instruction. 


ETYMOLOGY. 

Smith's  Complete  Etymology. 
Smith's  Condensed  Etymology. 

Containing  the  Anglo-Saxon,  French,  Dutch,  German,  Welsh,  Danish,  Gothic,  Swedish, 
Gaelic,  Italian,  Latin,  and  Greek  roots,  and  the  English  words  derived  therefrom 
accurately  spelled,  accented,  and  defined. 


From  HON.  JNO.  G.  McMvNN,  late  State 

Superintendent  of  Wisconsin. 

"  I  wish  every  teacher  in  the  country 
had  a  copy  of  this  work." 

From    PROF.  C.  H.  VERRILL,   Pa.    State 

Normal  School. 

uThe  Etymology  (Smith's)  which  we 
procured  of  you  we  like  much.  It  is  the 
best  work  for  the  class-rooin  we  have 
seen." 


From  PRIN.  WM.  F.  F  HELPS,  Minn.  Slate 
Normal. 

"The  book  is  superb— just  what  is 
needed  in  the  department  of  etymology 
and  spelling." 

From  HON.  EDWARD  BALLARD,  Supt.  oj 
Common  Schools,  State  of  Maine. 

''  The  author  has  furnished  a  manual  of 
singular  utility  for  its  purpose." 


DICTIONARY. 

Williams's  Dictionary  of  Synonymes  ; 

Or,  TOPICAL  LEXICON.  This  work  is  a  School  Dictionary,  an  Etymology,  a  compilation 
of  Synonymes,  and  a  manual  of  General  Information.  It  differs  from  the  ordinary  lexicon 
in  being  arranged  by  topics,  instead  of  the  letters  of  the  alphabet,  thus  realizing  the 
apparent  paradox  of  a  "  Readable  Dictionary."  An  unusuaJ'y  valuable  school-book. 

Kwong's  Dictionary  of  English  Phrases. 

With  Illustrative  Sentences,  collections  of  English  and  Chinese  Proverbs,  transla- 
tions of  Latin  and  French  Phrases,  historical  sketch  of  the  Chinese  Empire,  a  chrono- 
logical list  of  the  Chinese  Dynasties,  brief  biographical  sketches  of  Confucius  and 
of  Jesus,  and  complete  index.  By  Kwong  Ki  Chin,  late  Member  of  the  Chinese  Edu- 
cational Mission  \n  the  United  States,  and  formerly  principal  teacher  of  English  in  the 
Government  School  at  Shanghai,  China.  9(0piges.  Svo.  Cloth. 

From  the  Hartford  Courant :  "  The  volume  is  one  of  the  most  curious  and  interest- 
ing of  linguistic  works." 

From  the  New  York  Nation  :  "  It  will  amaze  the  sand-lot  gentry  to  bs  informed  that 
this  remarkable  work  will  supplement  our  English  dictionaries  even  for  native  Americans." 

10 


THE  NATIONAL   SERIES  Or   STANDARD  SCHOOL-BOOKS. 


ENGLISH    GRAMMAR. 


SILL'S  SYSTEM. 
Practical  Lessons  in  English. 

A  brief  course  in  Grammar  and  Composition.  By  J.  M.  B.  SILL.  This  beautiful 
hook,  by  a  distinguished  and  experienced  teacher,  at  once  adopted  for  exclusive  use 
in  the  State  of  Oregon  and  the  city  of  Detroit,  simply  releases  English  Grammar 
from  bondage  to  Latin  and  Greek  formulas.  Our  language  is  worthy  of  being  tanglit 
as  a  distinct  and  indepei.dent  science.  It  is  almost  destitute  of  inflections  and  yet 
capable  of  being  systematized,  and  its  study  may  certainly  be  simplified  if  treated  by 
itself  and  for  itsslf  alone.  Superintendent  SILL  has  cut  the  Gordian  knot  and  leads 
the  van  of  a  new  school  of  grammarians. 


CLARK'S    SYSTEM. 
Clark's  Easy  Lessons  in  Language 

Contains  illustrated  object-lessons  of  the  most  attractive  character,  and  is  couched 
in  language  freed  as  much  as  possible  from  the  dry  technicalities  of  the  science. 

Clark's  Brief  English  Grammar. 

Part  I.  is  adapted  to  youngest  learners,  and  the  whole  forms  a  complete  "  brief 
course  "  in  one  volume,  adequate  to  the  wants  of  the  common  school.  There  is  no- 
where published  a  superior  text-book  for  learning  thj  English  tongue  than  this. 

Clark's  Normal  Grammar. 

Designed  to  occupy  the  same  grade  as  the  author's  veteran  "Practical"  Grammar, 
though  the  latter  is  still  furnished  upon  order.  The  Normal  is  an  entirely  new  treatise. 
It  is  a  full  exposition  of  the  system  as  described  below,  with  all  the  most  recent  im- 
provements. Some  of  its  peculiarities  are, — a  happy  blending  of  SYNTTHESES  with 
ANALYSES  ;  thorough  criticisms  of  common  errors  in  the  use  of  our  language  ;  and 
important  improvements  in  the  syntax  of  sentences  and  of  phrases. 

Clark's  Key  to  the  Diagrams. 

Clark's  Analysis  of  the  English  Language. 

Clark's  Grammatical  Chart. 

The  theory  and  practice  of  teaching  grammar  in  American  schools  is  meeting  with  a 
thorough  revolution  from  the  use  of  this  system.  While  the  old  methods  offer  profi- 
ciency to  the  pupil  only  after  much  weary  plodding  and  dull  memorizing,  this  affords 
from  the  inception  the  advantage  of  practical  Object  Teaching,  addressing  the  eye  by 
means  of  illustrative  figures  ;  furnishes  association  to  the  memory,  its  most  powerful 
aid,  and  diverts  the  pupil  by  taxjng  his  ingenuity.  Teachers  who  are  using  Clark's 
Grammar  uniformly  testify  that  they  and  their  pupils  find  it  the  most  interesting  study 
of  the  school  course. 

Like  all  great  and  radical  improvements,  the  system  naturally  met  at  first  with  much 
unreasonable  opposition.  It  has  not  only  outlived  the  greater  part  of  this  opposition, 
bat  finds  many  of  its  warmest  admirers  among  those  who  could  not  at  first  tolerate  so 
radical  an  innovation.  All  it  wants  is  an  impartial  trial  to  convince  the  most  scep- 
tical of  its  merit.  No  one  who  has  fairly  and  intelligently  tested  it  in  the  school-room 
has  ever  been  known  to  go  back  to  the  old  method.  A  great  success  is  nlre-idy 
established,  and  it  is  easy  to  prophesy  that  the  day  is  not  far  distant  when  it  will  be 
the  only  system  of  teaching  English  Grammar.  As  the  SYSTEM  is  copyrighted,  no  other 
text-books  can  appropriate  this  obvious  and  great  improvement. 

Welch's  Analysis  of  the  English  Sentence. 

Remarkable  for  its  new  and  simple  classification,  its  method  of  treating  connectives, 
its  explanations  of  the  idioms  and  constructive  laws  of  the  language,  &c. 

11 


THE  NATIONAL   SERIES   OF   STANDARD   SCHOOL-BOOKS. 


GEOGRAPHY. 

MONTEITH'S  SYSTEM. 

TWO-BOOK  SERIES.      INDEPENDENT  COURSE. 

Elementary  Geography. 

Comprehensive  Geography  (with  103  maps). 

ESP"  These  volumes  are  not  revisions  of  old  works,  not  an  addition  to  any  series, 
but  are  entirely  new  productions,  — each  by  itself  complete,  independent,  compieheii- 
sive,  yet  simple,  brief,  cheap,  and  popular ;  or,  taken  together,  the  most  admirable 
"  series  "  ever  ottered  for  a  common-school  course.  They  present  the  following  features, 
skilfully  interwoven,  the  student  learning  all  about  one  country  at  a  time.  Always 
revised  to  date  of  printing. 

LOCAL  GEOGRAPHY.  —  Or,  the  Use  of  Maps.  Important  features  of  the  maps 
are  the  coloring  of  States  as  objects,  and  the  ingenious  system  for  laying  down  a  much 
larger  number  of  names  for  reference  than  are  lound  on  any  other  maps  of  same  size, 
and  without  crowdincr. 

PHYSICAL  GEOGRAPHY.— Or,  the  Natural  Features  of  the  Earth;  illus- 
trated by  the  original  and  striking  RELIEF  MAPS,  being  bird's-eye  views  or  photographic 
pictures  of  the  earth's  surface. 

DESCRIPTIVE  GEOGRAPHY.  — Including  the  Physical;  with  some  account 
of  Governments  and  Races,  Animals,  &e. 

HISTORICAL  GEOGRAPHY.  — Or,  a  brief  summary  of  the  salient  points  of 
history,  explaining  the  present  distribution  of  nations,  origin  of  geographical 
names,  &c. 

MATHEMATICAL  GEOGRAPHY.  —  Including  Astronomical,  which  describes 
the  Earth's  position  and  character  among  planets  ;  also  the  Zones,  Parallels,  &c. 

COMPARATIVE  GEOGRAPHY. —Or,  a  system  of  analogy,  connecting  new 
lessons  with  the  previous  ones.  Comparative  sizes  and  latitudes  are  shown  on  the 
margin  of  each  map,  and  all  countries  are  measured  in  the  "  frame  of  Kansas." 

TOPICAL  GEOGRAPHY. —  Consisting  of  questions  for  review,  and  testing 
the  student's  general  and  specific  knowledge  of  the  subject,  with  suggestions  for 
geographical  compositions. 

ANCIENT  GEOGRAPHY.  —  A  section  devoted  to  this  subject,  with  maps,  will 
be  appreciated  by  teachers.  It  is  seldom  taught  in  our  common  schools,  because  it 
has  heretofore  required  the  purchase  of  a  separate  book. 

GRAPHIC  GEOGRAPHY,  or  Map-Drawiug  by  Allen's  "Unit  of  Measure- 
ment" system  (now  almost  universally  recognized  as  without  a  rival),  is  introduced 
throughout  the  lessons,  and  not  as  an  appendix. 

CONSTRUCTIVE  GEOGRAPHY.  — Or,  Globe-Making.  With  each  book  a  set 
of  map  segments  is  furnished,  with  which  each  student  may  make  his  own  globe  by 
ibllowing  the  directions  given. 

RAILROAD  GEOGRAPHY.  —  With  a  grand  commercial  map  of  the  United 
States,  illustrating  steamer  and  railroad  routes  of  travel  in  the  United  States,  submarine 
telegraph  lines,  &c.  Also  a  "  Practical  Tour  in  Europe." 


MONTEITH     AND     McNALLY'S     SYSTEM. 

THREE  AND  FIVE  BOOKS.    NATIONAL  COURSE. 

Monteith's  First  Lessons  in  Geography. 
Monteith's  New  Manual  of  Geography. 
McNally's  System  of  Geography. 

The  new  edition  of  McNally's  Geography  is  now  ready,  rewritten  throughout  by 
James  Mouteitli  and  S.  C.  Frost.  In  its  new  dress,  printed  from  new  type,  and  illus- 
trated with  100  new  engravings,  it  is  the  latest,  most  attractive,  as  well  as  the  most 
thoroughly  practical  book  on  geography  extant. 

13 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

G  EOG  RAPH  Y  —  Continued. 

INTERMEDIATE   OR   ALTERNATE   VOLUMES    IN   THE   FIVE   BOOK   SERIES. 

Monteith's  Introduction  to  Geography. 
Monteith's  Physical  and  Political  Geography. 

x.  PRACTICAL  OBJECT-TEACHING.  —The  infant  scholar  is  first  introduced 
to  a  picture  whence  he  may  derive  notions  of  the  shape  of  the  earth,  the  phenomena  of 
day  and  night,  the  distribution  of  land  and  water,  and  the  great  natural  divisions, 
which  mere  words  would  fail  entirely  to  convey  to  the  untutored  mind.  Other  pictures 
follow  on  the  same  plan,  and  the  child's  mind  is  called  upon  to  grasp  no  idea  without 
the  aid  of  a  pictorial  illustration.  Carried  on  to  the  higher  books,  this  system  culmi- 
nates in  Physical  Geography,  where  such  matters  as  climates,  ocean  currents,  the 
winds,  peculiarities  of  the  earth's  crust,  clouds  and  rain,  are  pictorially  explained  and 
rendered  apparent  to  the  most  obtuse.  The  illustrations  used  for  this  purpose  belong 
to  the  highest  grade  of  art. 

2.  CLEAR,   BEAUTIFUL,   AND   CORRECT  MAPS.  —  In  the  lower  num- 
bers the  maps  avoid  unnecessary  detail,  while   respectively  progressive  and  affording 
the  pupil  new  matter  for  acquisition  each  time  he  approaches  in  the  constantly  en- 
larging circle  the  point  of  coincidence  with  previous  lessons  in  the  more  elementary 
books.     In  the  Physical  and   Political  Geography  the  maps  embrace  many  new  and 
striking  features.     One  of  the  most  effective  of  these  is  the  new  plan  for  displaying  on 
each  map  the  relative  sizes  of  countries  not  represented,  thus   obviating  much  confu- 
sion which  has  arisen  from  the  necessity  of  presenting  maps  in  the  same  atlas  drawn 
on  different  scales.     The  maps   of  "McNally"   have  long  been   celebrated   for  their 
superior  beauty  and  completeness.     This  is  the  only  school-book  in  which  the  attempt 
to  make  a  complete  atlas  also  clear  and  distinct,  has  been  successful.     The  map  coloring 
throughout  the  series  is  also  noticeable.     Delicate  and  subdued  tints  take  the  place  of 
the  startling  glare  of  inharmonious  colors  which  too  frequently  in  such  treatises  dazzle 
the  eyes,  distract  the  attention,  and  serve  to  overwhelm  the  names  of  towns  and  the 
natural  features  of  the  landscape. 

3.  THE   VARIETY   OF    MAP-EXERCISE. —Starting  each  time  from  a  dif- 
ferent basis,  the  pupil  in  many  instances   approaches  the  same  fact  no  less  than  six 
times,  thus  indelibly  impressing  it  upon  his  memory.     At  the  same  time,  this  system  is 
not  allowed  to  become  wearisome,  the  extent  of  exercise  on  each  subject  being  grad- 
uated by  its  relative  importance  or  difficulty  of  acquisition. 

4.  THE    CHARACTER    AND    ARRANGEMENT    OF    THE    DESCRIP- 
TIVE  TEXT.  — The  cream  of  the   science  has   been  carefully  culled,  unimportant 
matter  rejected,  elaboration  avoided,  and  a  brief  and  concise  manner  of  presentation 
cultivated.     The  orderly  consideration  of  topics  has  contributed  greatly  to  simplicity. 
Due  attention  is  paid  to  the  facts  in  history  and  astronomy  which  are  inseparably  con- 
nected with  and  important  to  the  proper  understanding  of  geography,  and  such  only 
are  admitted  on  any  terms.     In  a  word,  the  National  System  teaches  geography  as  a 
science,  pure,  simple,  and  exhaustive. 

5.  ALWAYS   UP  TO  THE  TIMES.  —  The  authors  of  these  books,  editorially 
speaking,  never  sleep.     No  change  occurs  in  the  boundaries  of  countries  or  of  counties, 
no  new  discovery  is  made,  or  railroad  built,  that  is  not  at  once  noted  and  recorded,  and 
the  next  edition  of  each  volume  carries  to  every  school-room  the  new  order  of  thinvs. 

6.  FORM    OF    THE   VOLUMES    AND    MECHANICAL    EXECUTIpN, 
—  The  maps  and  text  are  no  longer  unnaturally  divorced  in  accordance  with  the  time- 
honored  practice  of  making  text-books  on  this  subject  as  inconvenient  and  expensive  as 
possible.     On  the  contrary,  all  map  questions  are  to  be  found  on  the  page  opposite  tl.e 
-map  itself,  and  each  book  is  complete  in  one  volume.     The  mechanical  execution  is 

unrivalled.     Paper,  printing,  and  binding  are  everything  that  could  be  desired. 

7.  MAP-DRAWING.  —  In  1869  the  system  of  map-drawing  devised  by  Professor 
Jerome  Allen  was  secured  exclusively  for  this  series.     It  derives  its  claim  to  original- 
ity and  usefulness  from  the  introduction  of  a  fixed  unit  of  measurement  applicable  to 
every  map.     The  principles  being  so  few,  simple,  and  comprehensive,  the  subject  of 
map-drawing  is  relieved  of  all  practical  difficulty.    (In  Nos.  2,  2*,  and  3,  and  published 
separately.) 

8.  ANALOGOUS   OUTLINES. —At  the  same  time  with  map-drawing  was  also 
introduced  (in  No.  2)  a  new  and  ingenious  variety  of  Object  Lessons,  consisting  of  a 
comparison  of  the  outlines  of  countries  with  familiar  objects  pictorially  represented. 

14 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

GEOGRAPHY—  Continued. 

9.  SUPERIOR  GRADATION.  —  This  is  the  only  series  which  furnishes  an  avail- 
able volume  for  every  possible  class  in  graded  schools.  It  is  not  contemplated  that  a 
pupil  must  necessarily  go  through  every  volume  in  succession  to  attain  pn/i;:ieney. 
Uu  the  contrary,  two  will  suffice,  but  three  are  advised  ;  and,  if  the  course  will  admit, 
the  whole  series  should  be  pursued.  At  all  events,  the  books  are  at  hand  tor  selection) 
and  every  teacher,  of  every  grade,  can  find  among  them  one  exactly  suited  to  his  class. 
The  best  combination  for  those  who  wisli  to  abridge  the  course  consists  of  Nos.  1,  2, 
and  3  ;  or,  where  children  are  somewhat  advanced  in  other  studies  when  they  com- 
mence geography,  Nos.  1*,  2,  and  3.  Where  but  two  books  are  admissible,  Nos.  1*  and 
7  ,  or  Nos.  2  and  3,  are  recommended. 


A  SHEEP  RANCH  IN  MONTANA. 
[Specimen  Illustration  from  McNally's  New  Geography.] 


13 


THE  NATIONAL   SERIES   OF  STANDARD   CCHOOL-BOOKS. 

GEOGRAPHY  —  Continued. 

Monteith's  Physical  Geography. 

This  is  a  clear,  brief  statement  of  the  pnysical  attributes  of  the  earth  and  their  rela- 
tions to  the  heavens.  The  illustrations  ai.u  maps  are  numerous  and  helpful.  It  pro- 
vides full  instruction  in  this  important  branch  of  study  in  an  attractive  way  for  the 
youngest  scholars.  It  contains  i>4  pages  in  quarto  form. 


MAP-DRAWING. 

Monteith's  Map-Drawing  Made  Easy. 

A  neat  little  book  of  outlines  and  instructions,  giving  the  "corners  of  States"  in 
suitable  blanks,  so  that  maps  can  be  drawn  by  unskilful  hands  from  any  atlas  ;  with 
instructions  for  written  exercises  or  compositions  on  geographical  subjects,  and  com- 
parative geography. 

Monteith's  Manual  of  Map-Drawing  (Allen's  System). 

The  only  consistent  plan,  by  which  all  maps  are  drawn  on  one  scale.  By  its  use 
much  time  may  be  saved,  and  much  interest  and  accurate  knowledge  gained. 

Monteith's  Map-Drawing  and  Object  Lessons. 

The  last-named  treatise,  bound  with  Mr.  Monteith's  ingenious  system  for  commit- 
ting outlines  to  memory  by  means  of  pictures  of  living  creatures  and  familiar  objects. 
Thus,  South  America  resembles  a  dog's  head ;  Cuba,  a  lizard ;  Italy,  a  boot ;  France,  a 
coifee-pot  ;  Turkey,  a  turkey,  &c.,  &c. 

Monteith's  Colored  Blanks  for  Map-Drawing. 

A  new  aid  in  teaching  geography,  which  will  be  found  especially  useful  in  recitations, 
reviews,  and  examinations.  The  series  comprises  any  section  of  the  world  required. 

Monteith's  Map-Drawing  Scal^. 

A  ruler  of  wood,  graduated  to  the  "Allen  fixed  unit  of  measurement." 


WALL    MAPS. 

Monteith's  Pictorial  Chart  of  Geography. 

The  original  drawing  for  this  beautiful  and  instructive  chart  was  greatly  admired  in 
the  publisher's  "  exhibit  "  at  the  Centennial  Exhibition  of  1876.  It  is  a  picture  of  the 
earth's  surface  with  every  natural  feature  displayed,  teaching  also  physical  geography, 
and  especially  the  mutations  of  water.  Tke  uses  to  which  man  puts  the  earth  and  its 
treasures  and  forces,  as  Agriculture,  Mining,  Manufacturing,  Commerce,  and  Transpor- 
tation, are  also  graphically  portayed,  so  that  the  young  learner  gets  a  realistic  idea  of 
"the  world  we  live  in,"  which  weeks  of  book  study  might  fail  to  convey. 

Monteith's  School  Maps,  8  Numbers. 

-     The   "School   Series"  includes  the   Hemispheres   (2   maps),  United   States,   North 
America,  South  America,  Europe,  Asia,  Africa.     Price,  $2.50  each. 

Each  map  is  28x34  inches,  beautifully  colored,  has  the  names  all  laid  down,  and  is 
substantially  mounted  on  canvas  with  rollers. 

Monteith's  Grand  Maps,  8  Numbers. 

The  "Grand  Series"  includes  the  Hemispheres  (1  map).  North  America,  United 
States,  South  America,  Europe,  Asia,  Africa,  the  World  on  Mercator's  Projection,  and 
Physical  Map  of  the  World.  Price,  §5.00  each.  Size,  42  x  52  inches,  names  laid  down, 
colored,  mounted,  &c. 

Monteith's  Sunday-School  Maps. 

Including  a  map  of  Paul's  Travels  (§5.00),  one  of  Ancient  Canaan  (£3. 00),  and  Mod- 
ern Palestine  (-?3.00),  or  Palestine  and  Canaan  together  (.$5.00). 

16 


THE  NATIONAL   SERIES  OF  STANDARD  SCHOOL-BOOKS. 


MATHEMATICS 


DAVIES'S    COMPLETE 


Davies 
Davies 
Davies 


Davies'   Primary  Arithmetic. 
Davies'   Intellectual  Arithmetic. 

Elements  of  Written  Arithmetic. 

Practical  Arithmetic. 

University  Arithmetic. 

TWO-BOOK    SERIES. 

First  Book  in  Arithmetic,   Primary  and  Mental 
Complete  Arithmetic. 

ALGEBRA. 

Davies'   New  Elementary  Algebra. 
Davies'   University  Algebra. 
Davies'   New  Bourdon's  Algebra. 

GEOMETRY. 

Davies'  Elementary  Geometry  and  Trigonometry. 
Davies'   Legendre's  Geometry. 
Davies'  Analytical   Geometry   and  Calculus. 
Davies'   Descriptive  Geometry. 
Davies'   New  Calculus. 

MENSURATION. 

Davies'  Practical  Mathematics  and  Mensuration. 

Davies'  Elements  of   Surveying. 

Davies'   Shades,   Shadows,  and  Perspective. 

MATHEMATICAL   SCIENCE. 
Davies'   Grammar  of   Arithmetic. 
Davies'   Outlines  of   Mathematical  Science. 
Davies'   Nature  and  Utility  of   Mathematics. 
Davies'    Metric   System. 
Davies  &  Peck's  Dictionary  of  Mathematics. 

17 


THE   NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

DAVIES'S   NATIONAL   COURSE 
OF   MATHEMATICS. 

ITS    RECORD. 

In  claiming  for  this  series  the  first  place  among  American  text-books,  of  whatever 
class,  the  publishers  appeal  to  the  magnificent  record  which  its  volumes  have  earned 
during  the  tkirty-jhe  years  of  Dr.  Charles  Davies's  mathematical  labors.  The  unremit- 
ting exertions  of  a  l.fe-time  have  placed  the  modern  series  on  tlie  same  proud  eminence 
among  competitors  that  each  of  its  predecessors  had  successively  enjoyed  in  a  course  of 
constantly  improved  editions,  now  rounded  to  their  perfect  fruition,  —  for  it  seems 
almost  that  this  science  is  susceptible  of  no  further  demonstration. 

During  the  period  alluded  to,  many  authors  and  editors  in  this  department  h?ve 
started  into  public  notice,  and,  by  borrowing  ideas  and  processes  original  with  Dr.  Davies, 
have  enjoyed  a  brief  popularity,  but  are  now  almost  unknown.  Many  of  the  series  of 
to-day,  built  upon  a  similar  basis,  and  described  as  "  modern  books,"  are  destined  to  a 
similar  fate ;  while  the  most  far-seeing  eye  will  find  it  diflicult  to  fix  the  time,  on  the 
basis  of  any  data  afforded  by  their  past  history,  when  these  books  will  cease  to  increase 
and  prosper,  and  fix  a  still  firmer  hold  on  the  affection  of  every  educated  American. 

One  cause  of  this  unparalleled  popularity  is  found  in  the  fact  that  the  eiiterfin.se  of  the 
author  did  not  cease  with  the  original  completion  of  his  books.  Always  a  praciicai 
teacher,  he  has  incorporated  in  his  text-books  from  time  to  time  tlie  ad vantages  or  every 
improvement  in  method*  of  teaching,  and  every  advance  in  science.  During  ail  the 
years  in  which  he  has  been  laboring  he  constantly  submitted  his  own  theories  and  those 
of  others  to  the  practical  test  of  the  class-room,  approving,  rejecting,  or  modify  ing 
them  as  the  experience  thus  obtained  might  suggest.  In  this  way  he  has  been  aiile 
to  produce  an  almost  perfect  series  of  class-books,  in  which  every  department  of 
mathematics  has  received  minute  and  exhaustive  attention. 

Upon  tlie  death  of  Dr.  Davies,  which  took  place  in  1876,  his  work  was  immediately 
taken  up  by  his  former  pupil  and  mathematical  associate  of  many  years,  Prof.  W.  G. 
Peck,  LL.D.,  of  Columbia  College.  By  him,  with  Prof.  J.  H.  Van  Amringe,  of  Columbia 
College,  the  original  series  is  kept  carefully  revised  and  up  to  the  times. 


DAVIES'S  SYSTEM  is  THE  ACKNOWLEDGED  NATIONAL  STANDARD  FOR  THE  UNITED 
STATES,  for  the  following  reasons  :  — 

1st.  It  is  the  basis  of  instruction  in  the  great  national  schools  at  West  Point  and 
Annapolis. 

2d.    It  has  received  the  quasi  indorsement  of  the  National  Congress. 

3d.     It  is  exclusively  used  in  the  public  schools  of  the  National  Capital. 

4th.  The  officials  of  the  Government  use  it  as  authority  in  all  cases  involving  mathe- 
matical questions. 

5th.  Our  great  soldiers  and  sailors  commanding  the  national  armies  and  navies  were 
educated  in  this  system.  So  have  been  a  majority  of  eminent  scientists  in  this  country. 
All  these  refer  to  "  Davies  "  as  authority. 

6th.  A  larger  number  of  American  citizens  have  received  their  education  from  this 
than  from  any  other  series. 

7th.  The  scries  has  a  larger  circulation  throughout  the  whole  country  than  any  other, 
being  extensive;/!/  used  in  every  State  in  the  Union. 

It 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 
DAVIES     AND     PECK'S     ARITHMETICS. 

OPTIONAL   OR   CONSECUTIVE, 

The  best  thoughts  of  these  two  illustrious  mathematicians  are  combined  in  (he 
Following  beautiful  works,  which  are  the  natural  successors  of  Davies's  Arithmetics, 
sumptuously  printed,  and  bound  iu  crimson,  green,  and  gold:  — 

Davies  and  Peck's  Brief  Arithmetic. 

Also  called  the  "  Elementary  Arithmetic."  It  is  the  shortest  presentation  of  the  sub- 
ject, and  is  ndiqiiatt:  for  all  grades  in  common  schools,  being  a  thorough  introduction  to 
practical  lite,  except  for  the  specialist. 

At  h'rst  the  authors  play  with  the  little  learner  for  a  few  lessons,  by  object-teaching 
and  kindred  allurements  ;  but  he  soon  begins  to  realize  that  study  is  earnest,  as  he 
becomes  familiar  with  the  simpler  operations,  and  is  delighted  to  find  himself  master  of 
important  resuKs. 

The  second  part  reviews  the  Fundamental  Operations  on  a  scale  proportioned  iu 
the  enlarged  intelligence  of  the  learner.  It  establishes  the  General  Principles  and 
Properties  of  Numbers,  and  then  proceeds  to  Fractions.  Currency  and  the  Metric 
(System  are  fully  treated  in  connection  with  Decimals.  Compound  Numbers  and  Re- 
duction follow,  and  finally  Percentage  with  all  its  varied  applications. 

An  Index  of  words  and  principles  concludes  the  book,  for  which  every  scholar  and 
most  teachers  will  be  grateful.  How  much  time  has  been  spent  in  searching  for  a  half- 
forgotten  definition  or  principle  in  a  former  lesson  ! 

Davies  and  Peck's  Complete  Arithmetic. 

This  work  c  :rtainly  deserves  its  name  in  the  best  sense.  Though  complete,  it  is  not, 
like  most  others  which  hear  the  same  title,  cumbersome.  These  authors  excel  in  clear, 
lucid  demonstrations,  teaching  tl.e  science  pure  and  simple,  yet  not  ignoring  convenient 
methods  ;:i.d  practical  applications. 

For  turning  out  a  thorough  business  man  no  other  work  is  so  well  adapted.  He  will 
have  a  clear  comprehension,  of  the  science  as  a  whole,  and  a  working  acquaintance 
with  detai.s  which  must  serve  him  well  in  all  emergencies.  Distinguishing  features  of 
the  book  are  the  logical  progression  of  the  subjects  and  the  great  variety  of  practical 
problems,  not.  puzzle*,  which  are  beneath  the  dignity  of  educational  science.  A  clear- 
minded  critic  Las  said  of  Dr.  Peck's  work  that  it  is  free  from  that  juggling  with 
numbers  which  fome  authors  falsely  call  "  Analysis."  A  series  of  Tables  for  converting 
ordinal  y  weights  ahd  measures  into  the  Metric  .System  appear  in  the  later  editions. 


PECK'S    ARITHMETICS. 
Peck's  First  Lessons  in  Numbers. 

This  book  begins  with  pictorial  illustrations,  and  unfolds  gradually  the  science  of 
numbers.  It  noticeably  simplifies  the  subject  by  developing  the  principles  of  addition 
and  subtraction  simultaneously  ;  as  it  does,  also,  those  of  multiplication  and  division. 

Peck's  Manual  of  Arithmetic. 

This  hook  is  designed  especially  'or  those  who  seek  sufficient  instruction  to  carry 
them  successfully  through  practical  life,  but  have  not  time  for  extended  study. 

Peck's  Complete  Arithmetic. 

This  completes  the  series  but  is  a  much  briefer  book  than  most  of  the  complete 
arithmetics,  and  is  recommended  not  only  for  what  it  contains,  but  also  for  what  is 
omitted. 

It  may  be  said  of  Dr.  Peck's  books  more  truly  than  of  any  other  series  published,  that 
they  are  clear  and  simple  in  definition  and  rule,  and  that  superfluous  matter  of  every 
Kind  has  been  faithfully  eliminated,  thus  magnifying  the  working  value  of  the  book 
«ud  saving  unnecessary  expense  of  time  and  labor. 

19 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


BARNES'S  NEW  MATHEMATICS. 

In  this  series  JOSEPH  FICKLIN,  Ph.  D.,  Proi'essor  of  Mathematics  and  Astronomy 
in  the  University  of  Missouri,  lias  combined  all  the  best  and  latest  results  of  practical 
and  experimental  teaching  of  arithmetic  with  the  assistance  of  many  distinguished 
mathematical  authors. 


Barnes's  Elementary  Arithmetic. 
Barnes's  National  Arithmetic. 

These  two  works  constitute  a  complete  arithmetical  course  in  two  books. 

They  meet  the  demand  for  text-books  that  will  help  students  to  acquire  the  greatest 
amount  of  useful  and  practical  knowledge  of  Arithmetic  by  the  smallest  expenditure  of 
time,  labor,  and  money.  Nearly  every  topic  in  Written  Arithmetic  is  introduced,  and  its 
principles  illustrated,  by  exercises  in  Oral  Arithmetic.  The  free  use  of  Equations  ;  the 
concise  method  of  combining  and  treating  Properties  of  Numbers;  the  treatment  of 
Multiplication  and  Division  of  Fractions  in  two  cases,  and  then  reduceO-to  one;  Can- 
cellation by  the  use  of  the  vertical  line,  especially  in  Fractions,  Interest,  and  Proportion  ; 
the  brief,  simple,  and  greatly  superior  method  of  working  Partial  Payments  by  the 
41  Time  Table  "  and  Cancellation  ;  the  substitution  of  formulas  to  a  great  extent  for 
rules;  the  full  and  practical  treatment  of  the  Metric  System,  &c.,  indicate  their  com- 
pleteness. A  variety  of  methods  and  processes  for  the  same  topic,  which  deprive  the 
pupil  of  the  great  benefit  of  doing  a  part  of  the  thinking  and  labor  for  himself,  have 
beeu  discarded.  The  statement  of  principles,  definitions,  rules,  &c.,  is  brief  and  simple. 
The  illustrations  and  methods  are  explicit,  direct,  and  practical.  The  great  number 
and  variety  of  Examples  embody  the  actual  business  of  the  day.  The  very  large 
amount  of  matter  condensed  in  so  small  a  compass  has  beeu  accomplished  by  econo- 
mizing every  line  of  space,  by  rejecting  superfluous  matter  and  obsolete  terms,  and  by 
avoiding  the  repetition  of  analyses,  explanations,  and  operations  in  the  advanced  topics 
which  have  been  used  in  the  more  elementary  parts  of  these  books. 

AUXJLI  ARIES. 

For  use  in  district  schools,  and  for  supplying  a  text-book  in  advanced  work  for 
{•lasses  having  finished  the  course  as  given  in  the  ordinary  Practical  Arithmetics,  the 
National  Arithmetic  has  been  divided  and  boutd  separately,  as  follows  :  — 

Barnes's  Practical  Arithmetic. 
'    liarnes's  Advanced  Arithmetic. 

In  many  schools  there  are  classes  that  for  various  reasons  never  reach  beyond 
Percentage.  It  is  just  such  cases  where  Barnes's  Practical  Arithmetic  will  answer  a 
good  purpose,  at  a  price  to  the  jwpil  much  less  than  to  buy  the  complete  book.  On  the 
other  hand,  classes  having  finished  the  ordinary  Practical  Arithmetic  can  proceed 
with  the  higher  course  by  using  Barnes's  Advanced  Arithmetic. 

For  primary  schools  requiring  simply  a  table  Ixwk,  and  the  earliest  rudiments 
forcibly  presented  through  object-teaching  and  copious  illustiations,  we  have 
prepared 

Barnes's  First  Lessons  in  Arithmetic, 

which  begins  with  the  most  elementary  notions  of  numbers,  and  proceeds,  by  simple 
steps,  to  develop  all  the  fundamental  principles  of  Arithmetic. 


Barnes's  Elements  of  Algebra. 

This  work,  as  its  title  indicates,  is  elementary  in  its  character  and  suitable  for  use, 
(1)  in  such  public  schools  as  give  instruction  in  the  Elements  of  Algebra  :  (2)  in  institu- 
tions of  learning  whose  courses  of  study  do  not  include  Higher  Algebra  ;  (3)  in  schools 
whose  object  is  to  prepare  students  for  entrance  into  our  colleges  and  universities. 
This  book  will  also  meet  the  wants  of  students  of  Physics  who  require  some  knowledge  ot 

20 


THE  NATWAL   SERIES  OF  STANDARD  SCHOOL-BOOKS. 


Algebra.  The  student's  progress  in  Algebra  depends  very  largely  upon  the  proper  treat- 
ment of  the  four  Fundamental  Operations*  The  terms  Addition,  Subtraction,  Multiplication, 
and  Division  in  Algebra  have  a  wider  meaning  than  in  Arithmetic,  and  these  operations 
have  been  so  denned  as  to  include  their  arithmetical  meaning  ;  so  that  the  beginner 
is  sinrply  called  upon  to  enlarge  his  views  of  those  fundamental  operations.  Much 
attention  has  been  given  to  the  explanation  of  the  negative  sign,  in  order  to  remove  the 
well-known  difficulties  in  the  use  and  interpretation  of  that  sign.  Special  attention  is 
here  called  to  "  A  Short  Method  of  Removing  Symbols  of  Aggregation,"  Art.  76.  On 
account  of  their  importance,  the  subjects  of  Factoring,  Greatest  Cowman  Dirisor,  and 
Least  Common  Multiple  have  been  treated  at  greater  length  than  Is  usual  in  elementary 
works.  In  the  treatment  of  Fractions,  a  method  is  used  which  is  quite  simple,  and', 
at  tl/e  same  time,  more  general  than  that  usually  employed.  In  connection  with  R<idic«l 
Quantities  the  roots  are  expressed  by  fractional  exponents,  for  the  principles  and  rules 
applicable  to  integral  exponents  may  then  be  used  without  modification.  The  Equation 
is  made  the  chief  subject  of  thought  in  this  work.  It  is  defined  near  the  beginning, 
and  used  extensively  in  every  chapter.  In  addition  to  this,  four  chapters  are  devoted 
exclusively  to  the  subject  of  Equations.  All  Proportions  are  equations,  and  in  their 
treatment  as  such  all  the  difficulty  commonly  connected  with  the  subject  of  Proportion 
disappears.  The  chapter  on  Logarithms  will  doubtless  be  acceptable  to  many  teachers 
who  do  not  require  the  student  to  master  Higher  Algebra  before  entering  upon  the 
study  of  Trigonometry. 


HIGHER     MATHEMATICS. 
Peck's  Manual  of  Algebra. 

Bringing  the  methods  of  Bourdon  within  the  range  of  the  Academic  Course. 

Peck's  Manual  of  Geometry. 

By  a  method  purely  practical,  and  unembarrassed  by  the  details  which  rather  confuse 
than  simplify  science. 

Peck's  Practical  Calculus. 
Peck's  Analytical  Geometry. 
Peck's  Elementary  Mechanics. 
Peck's  Mechanics,  with  Calculus. 

The  briefest  treatises  on  these  subjects  now  published.  Adopted  by  the  great  Univer- 
sities :  Yale,  Harvard,  Columbia,  Princeton,  Cornell,  &c. 

Macnie's  Algebraical  Equations. 

Serving  as  a  complement  to  the  more  advanced  treatises  on  Algebra,  giving  special 
attention  to  the  analysis  and  solution  of  equations  with  numerical  coefficients. 

Church's  Elements  of  Calculus. 

Church's  Analytical  Geometry. 

Church's  Descriptive  Geometry.     With  plates.     %  vols. 

These  volumes  constitute  the  "West  Point  Course  "in  their  several  deparments. 
Prof.  Church  was  long  the  eminent  professor  of  mathematics  at  West  Point  Military 
Academy,  and  his  works  are  standard  in  all  the  leading  colleges. 

Courtenay's  Elements  of  Calculus. 

A  standard  work  of  the  very  highest  grade,  presenting  the  most  elaborate  attainable 
survey  of  the  subject. 

Hackley's  Trigonometry. 

With  applications  to  Navigation  and  Surveying,  Nautical  and  Practical  Geometry, 
and  Geodesy. 

21 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


GENERAL    HISTORY. 

Monteith's  Youth's  History  of  the  United  States. 

A  History  of  the  United  States  for  beginners.  It  is  arranged  upon  the  catechetical  plan, 
with  illustrative  maps  and  engravings,  re  view  questions,  dates  in  parentheses  (that  their 
study  may  be  optional  with  the  younger  class  of  learners),  and  interesting  biographical 
sketches  of  all  persons  who  have  been  prominently  identified  with  the  history  of  our 
country. 

Willard's  United   States.      School  and  University  Editions. 

The  plan  of  this  standard  work  is  chronologically  exhibited  in  front  of  the  titlepage. 
The  maps  and  sketches  are  found  useful  assistants  to  the  memory  ;  and  dates,  usually 
so  difficult  to  remember,  are  so  systematically  arranged  as  in  a  great  degree  to  obviate 
the  difficulty.  Candor,  impartiality,  and  accuracy  are  the  distinguishing  features  of 
the  narrative  portion. 

Willard's  Universal  History.     New  Edition. 

The  most  valuable  features  of  the  "  United  States  "  are  reproduced  in  this.  The 
peculiarities  of  the  work  are  its  great  conciseness  and  the  prominence  given  to  the 
chronological  order  of  events.  The  margin  marks  each  successive  era  with  great  dis- 
tinctness, so  that  the  pupil  retains  not  only  the  event  but  its  time,  and  thus  fixes  the 
order  of  history  firmly  and  usefully  in  his  mind.  Mrs.  Willard's  books  are  constantly 
revised,  and  at  all  times  written  up  to  embrace  important  historical  events  of  recent 
date.  Professor  Arthur  Oilman  has  edited  the  last  twenty-five  years  to  1882. 

Lancaster's  English  History. 

By  the  Master  of  the  Stoughton  Grammar  School,  Boston.  The  most  practical  of  the 
"  brief  books."  Though  short,  it  is  not  a  bare  and  uninteresting  outline,  but  contains 
enough  of  explanation  and  detai  1  to  make  intelligible  the  cause  and  effect  of  events. 
Their  relations  to  the  history  and  development  of  the  American  people  is  made  specially 
prominent. 

Willis's  Historical  Reader. 

Being  Collier's  Great  Events  of  History  adapted  to  American  schools.  This  rare 
epitome  of  general  history,  remarkable  for  its  charming  style  and  judicious  selection  of 
events  on  which  the  destinies  of  nations  have  turned,  has  been  skilfully  manipulated 
by  Professor  Willis,  with  as  few  changes  as  would  bring  the  United  States  into  its  proper 
position  in  the  historical  perspective.  As  reader  or  text-book  it  has  few  equals  and  no  . 
superior. 

Berard's  History  of  England. 

By  an  authoress  well  known  for  the  success  of  her  History  of  the  United  States. 
The  social  life  of  the  English  people  is  felicitously  interwoven,  as  in  fact,  with  the  civil 
and  military  transactions  of  the  realm. 

Ricord's  History  of  Rome. 

Possesses  the  charm  of  an  attractive  romance.  The  fables  with  which  this  history 
abounds  are  introduced  in  such  a  way  as  not  to  deceive  the  inexperienced,  while  adding 
materially  to  the  value  of  the  work  as  a  reliable  index  to  the  character  and  institutions, 
as  well  as  the  historv  of  the  Roman  people. 

22 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

HISTORY  —  Continued. 

Hanna's  Bible  History. 

The  only  compendium  of  Bible  narrative  which  affords  a  connected  and  chronological 
view  of  the  important  events  there  recorded,  divested  of  all  superfluous  detail. 

Summary  of  History ;  American,  French,  and  English. 

A  well-proportioned  outline  of  leading  events,  condensing  the  substance  of  the  more 
extensive  text-books  in  common  use  into  a  series  of  statements  so  brief,  that  every 
word  may  be  committed  to  memory,  and  yet  so  comprehensive  that  it  presents  an 
accurate  though  general  view  of  the  whole  continuous  life  of  nations. 

Marsh's  Ecclesiastical  History. 

Affording  the  History  of  the  Church  in  all  ages,  with  accounts  of  the  pagan  world 
during  the  biblical  periods,  and  the  character,  rise,  and  progress  of  all  religions,  as  well 
as  the  various  sects  of  the  worshippers  of  Christ.  The  work  is  entirely  non-sectarian, 
though  strictly  catholic.  A  separate  volume  contains  carefully  prepared  questions  for 
class  use. 

Mill's  History  of  the  Ancient  Hebrews. 

With  valuable  Chronological  Charts,  prepared  by  Professor  Edwards  of  N.  Y.  This 
is  a  succinct  account  of  the  chosen  people  of  God  to  the  time  of  the  destruction  of 
Jerusalem.  Complete  in  one  volume. 

Topical  History  Chart  Book. 

By  Miss  Ida  P.  Whitcomb.  To  be  used  in  connection  with  any  History,  Ancient  or 
Modern,  instead  of  the  ordinary  blank  book  for  summary.  It  embodies  the  names  of 
cnntfini>i>i-iu-ii  1-nlci-n  from  the  earliest  to  the  present  time,  with  blanks  under  each,  in 
which  the  pupil  may  write  the  summary  of  the  life  of  the  ruler. 

Oilman's  First  Steps  in  General  History. 

A  "suggestive  outline"  of  rare  compactness.  Each  country  is  treated  by  itself,  and 
the  United  States  receive  special  attention.  Frequent  maps,  contemporary  events  in 
tables,  references  to  standard  works  for  fuller  details,  and  a  minute  Index' constitute 
the  "  Illustrative  Apparatus.'1  From  no  other  work  that  we  know  of  can  so  succinct  a 
view  of  the  world's  history  be  obtained.  Considering  the  necessary  limitation  of  space, 
the  style  is  surprisingly  vivid,  and  at  times  even  ornate.  In  all  respects  a  charming, 
though  not  the  less  practical,  text-book. 

Baker's  Brief  History  of  Texas. 
Dimitry's  History  of  Louisana. 
Alison's  Napoleon  First. 

The  history  of  Europe  from  1788  to  1S15.  By  Archibald  Alison.  Abridged  by  Edward 
S.  Gould.  One  vol.,  Svo,  with  appendix,  questions,  and  maps.  550  pages. 

Lord's  Points  of  History. 

The  salient  points  in  the  history  of  the  world  arranged  catechetically  for  class  use  or 
for  review  and  examination  of  teacher  or  pupil.  By  John  Lord,  LL.D.  12mo,  300 

pages. 

Carrington's  Battle  Maps  and  Charts  of  the  American 
Revolution. 

Topographical  Maps  and  Chronological  Charts  of  every  battle,  with  3  steel  portraits 
of  Washington.  Svo,  cloth. 

Condit's  History  of  the   English  Bible. 

For  theological  and  historical  students  this  book  has  an  intrinsic  value.  It  gives  the 
history  of  all  the  English  translations  down  to  the  present  time,  together  with  a  careful 
review  of  their  influence  upon  English  literature  and  language. 

23 


THE  NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 


BARNES'S   ONE-TERM    HISTORY 
SERIES. 


A    Brief     History    of    the    United 
States. 

This  is  probably  the  MOST  ORIGINAL  SCHOOL-BOOK  pub- 
lished for  many  years,  in  any  department.  A  few  of  it? 
claims  are  the  following  :  — 

1.  Brevity.  —  The  text  is  complete  for  grammar  school 
or  intermediate  classes,  in   290   12im>  pages,  large   type. 
It  may  readily  be  completed,  if  desired,  in  one  term  of 
study. 

2.  Comprehensiveness.  —  Though  so  brief,  this  book 
contains  the  pith  of  all  the  wearying  contents  of  the  larger 
manuals,  and  a  great  deal  more  than  the  memory  usually 
retains  I'roin  the  latter. 

3.  Interest  has  been  a  prime  consideration.     Small 

books  have  heretofore  been  bare,  full  of  dry  statistics,   unattractive.    This  one  is 
charmingly  written,  replete  with  anecdote,  and  brilliant  with  illustration. 

4.  Proportion  of  Events.  —  It  is  remarkable  for  the  discrimination  with  which 
the  different  portions  of  our  history  are  presented  according  to  their  importance.     Thus 
the  older  works,  being  already  large  books  when  the  Civil  War  took  place,  give  it  less 
space  than  that  accorded  to  the  Revolution. 

5.  Arrangement.  —  In  six  epochs,  entitled  respectively,  Discovery  and  Settlement, 
the  Colonies,  the  Revolution,  Growth  of  States,  the  Civil  War.  and  Current  Events. 

6.  Catch  Words.  —  Each  paragraph  is  preceded  by  its  leading  thought  in  promi- 
nent type,  standing  in  the  student's  mind  for  the  whole  paragraph. 

7:- Key  Notes.  —  Analogous  with  this  is  the  idea  of  grouping  battles,  £c..,  about 
some  central  event,  which  relieves  the  sameness  so  common  in  such  descriptions,  and 
renders  each  distinct  by  some  striking  peculiarity  of  its  own. 

8.  Foot-Notes. — These  are  crowded  with  interesting  matter  that  is  not  strictly  a 
part  of  history  proper.     They  may  be  learned  or  not,  at  pleasure.     They  are  certain 
in  any  event  to  be  read. 

9.  Biogiaphies  of  all  the  leading  characters  are  given  in  full  in  foot-notes. 

10.  Maps.  —  Elegant  and  distinct  maps  from  engravings  on  copper-plate,  and  beauti- 
fully colored,  precede  each  epoch,  and  contain  all  the  places  named. 

n.  Questions  are  at  the  back  of  the  book,  to  compel  a  more  independent  use  of  the 
text.  Both  text  and  questions  are  so  worded  that  the  pupil  must  give  intelligent 
answers  IN  HIS  OWN  WORDS.  "  Yes"  and  "No  "  will  not  do. 


24 


THE  NATIONAL   SERIES  OF  STANDARD  SCHOOL-COOKS. 


HISTORY  —  Continued. 

12.  Historical  Recreations.  — These  are  additional  questions  to  test  the  student's 
knowledge,  in   review,  ;is  :    "What  trees  are  celebrated    in  our  history?"     "When 
did  a  Jog  save  our  army?"     "What  Presidents  died  in  office?"     "When  was  the 
Mississippi  our  western    boundary?"     "Who  said,    'I  would  rather  be  right  than 
President '  ?  "  &c. 

13.  The  Illustrations,  about  seventy  in  number,  are  the  work  of  our  best  artists 
and  engravers,  produced  at  great  expense.     They  are  vivid  and  interesting,  and  mostly 
upon  subjects  never  before  illustrated  in  a  school-book. 

14  Dates  —Only  the  leading  dates  are  given  in  the  text,  and  these  are  so  associated 
as  to  assist  the  memory,  but  at  the  head  of  each  page  is  the  date  of  the  event  imt 
mentioned,  and  at  the  close  of  each  epoch  a  summary  of  events  and  dates. 

15.  The  Philosophy  of    History  is   studiously  exhibited,  the   causes  and  effects 
of  events  being  distinctly  traced  and  their  inter-connection  shown. 

16.  Impartiality.  —  All  sectional,   partisan,  or  denominational  views  are  avoided. 
Facts  are  stated  after  a  careful  comparison  of  all  authorities  without  the  least  prejudice 
or  favor. 

17.  Index.  —  A  verbal  index  at  the  close  of  the  book  perfects  it  as  a  work  of  reference. 
It  will  be  observed  that  the  above  are  all  particulars  in  which  School  Histories  have 

been  signally  defective,  or  altogether  wanting.     Many  otlicr  claims  to  favor  it  shares  in 
common  with  its  predecessors. 


TESTIMONIALS. 


From   PROF.   WM.    F.   ALLEN,    Slate   Uni- 
versity of  H'isconsin. 

"Two  features  that  I  like  re.ry  much 
are  the  anecdotes  at  the  foot  of  the  page 
and  the  'Historical  Recreations'  in  the 
Appendix.  The  latter,  I  think,  is  quite 
a  n-w  feature,  and  the  other  is  very  well 
executed." 

From  HON.  NEWTON  BATEMAN,  Superin- 
tendent Public  Instruction,  Illinois. 
"  Barnes's  One-Term  History  of  the 
United  States  is  an  exceedingly  attrac- 
tive and  spirited  little  book.  Its  claim 
to  several  new  and  valuable  features  seems 
well  founded.  Under  the  form  of  six  well- 
defined  epochs,  the  history  of  the  United 
States  is  traced  tersely,  yet  pithily,  from 
the  earliest  times  to  the  present  day.  A 
good  map  precedes  each  epoch,  whereby 
the  history  and  geography  of  the  period 
may  be  studied  together,  ns  tttry  nlwnys 
sfionl  I.  be.  The  syllabus  of  each  paragraph 
is  made  to  stand  in  such  bold  relief,  by 
the  use  of  large,  heavy  type,  as  to  be  of 
much  mnemonic  value  to  the  student.  The 
book  is  written  in  a  sprightly  and  pi- 

Suant  style,  the  interest  never  llag.-'ing 
•om  beginning  to  end, —  a  rare  and  diffi- 
cult achievement  in  works  of  this  kind." 

From   HON.   ABNER  J.    PHIPPS,    Sipe rill- 
ten  lent  S-hools,  Lwiston,  Maine,. 
4-  Barnes's  History  of  the  United  States 


has  been  used  for  severnl  years  in  the 
Lewiston  schools,  and  has  proved  a  very 
satisfactory  work.  I  have  examined  the 
new  edition  of  it." 

From  HON.  R.  K.  BUCHET/L,  City  Superin- 
tendent S  hoofs,  L-nicaster,  Pa. 

"  It  is  the  best  history  of  the  kind  I  have 
ever  seen.'' 

From    T.    J.    CHARLTON,    Superintendent 

Public   Schools,    Vincmn-s,  In  I. 
"  We  have  used  it  here  for  six  years, 
and  it  has  given  almost  per.ect  satisfac- 
tion. .  .  .  The  notes  in  fine  print  at  the 
bottom  of  the  pages  are  of  especial  value." 

From  PROF.   WM.    A.    MOWRY,  E.  ,j-  C. 

School,    Providence,    Jt.   /. 

"  Permit  me  to  express  my  high  appre- 
ciation of  your  book.  I  wish  all  text- 
books for  the  young  had  equal  merit." 

From  HON.  A.  M.  K  El  LEY,  City  Attnrneij, 
Lute  Mayor,  and  President  of  the  School 
Board,  City  of  RichmonJ,  Va. 
"  I  do  not  hesitate  to  volunteer  to  you 
the  opinion  that  Barnes  's  History  is  en- 
titled to  the  preference   in  almost  every 
respect  that  distinguishes  a  good  school- 
book.  .  .  .  The  narrative  generally  exhibits 
the  temper  of  the  jud^e  ;  rarely,  if  ever, 
of  the  advocate." 


25 


THE  NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 


A  Brief  History  of  An- 
cient Peoples. 

With  an  account  of  their  monuments, 
literature,  and  manners.  340  pages. 
12mo.  Profusely  illustrated. 

In  this  work 'the  political  history, 
which  occupies  nearly,  if  not  all, 
the  ordinary  school  text,  is  condensed 
to  the  salient  and  essential  facts,  in 
order  to  give  room  for  a  clear  outline 
of  the  literature,  religion,  architecture, 
character,  habits,  &o.,  of  each  nation. 
Surely  it  is  as  important  to  know  some- 
tking'abmit  Plato  as  all  about  Caesar, 
and  to  learn  how  the  ancients  wrote 
their  books  as  how  they  fought  their 
battles. 

The  chapters  on  Manners  and  Cus- 
toms and  the  Scenes  in  Real  Life  repre- 
sent the  people  of  history  as  men  and 
women  subject  to  the  same  wants,  hopes 
and  fears  as  ourselves,  and  so  bring  the  distant  past  near  to  us.  The  Scenes,  which  are 
intended  only  for  reading,  are  the  result  of  a  careful  study  of  the  unequalled  collections  of 
monuments  in  the  London  and  Berlin  Museums,  of  the  ruins  in  Rome  and  Pompeii,  and 
of  the  latest  authorities  on  the  domestic  life  of  ancient  peoples.  Though  intentionally 
written  in  a  semi-romantic  style,  they  are  accurate  pictures  of  what  might  have  occurred 
and  some  of  them  are  simple  transcriptions  of  the  details  sculptured  in  Assyrian 
alabaster  or  painted  on  Egyptian  walls. 

26 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 


HISTORY  —  Continued. 

The  extracts  made  from  the  sacred  books  of  the  East  are  not  specimens  of  their  style 
and  teachings,  but  only  gems  selected  often  from  a  mass  of  matter,  much  of  which  would 
be  absurd,  ineaningless,  and  even  revolting.  It  has  not  seemed  best  to  cumber  a  booic 
like  this  with  selections  conveying  no  moral  lesson. 

Ihe  numerous  cross-references,  the  abundant  dates  in  parenthesis, the  pronunciation 
of  the  names  in  the  Index,  the  choice  reading  references  at  the  close  of  each  general 
subject,  and  the  novel  Historical  Recreations  in  the  Appendix,  will  be  of  service  to 
teacher  and  pupil  alike. 

Though  designed  primarily  for  a  text-book,  a  large  class  of  persons  — general  readers, 
who  desire  to  know  something  about  the  progress  of  historic  criticism  and  ,ne  recent 
discoveries  made  among  the  resurrected  monuments  of  the  East,  but  have  no  leisure  to 
read  the  ponderous  volumes  of  Brugsch,  Layard,  Grote,  Mommseu,  and  lime —  will  lind 
tli is  volume  just  what  they  need. 


Frnm  HOMER  B.    SPRAGUE,  Heal  Master 
Girls'  II/f/h  Schod,  H'e.st  Newton  St.,  Bos- 
t-n,M,ss. 
"  I  Vug  to  recommend  in  strong  terms 

the    adoption    of    Barnes's    'History    of 


Ancient  Peoples'  as  a  text-book.  It  is 
about  as  nearly  perfect  as  could  be 
hoped  for.  The  adoption  would  give 
great  relish  to  the  study  of  Ancient 
History." 


HE  Brief  History  of  France. 


By  the  author  of  the  "  Brhf  United  States," 
^r  with  all  the  attractive  features  of  that  popu- 
lar work  (which  see)  and  new  ones  of  its  own. 

It  is  believed  ti;at  the  History  of  France 
has  never  before  been  presented  in  such 
brief  compass,  and  this  is  effected  without 
sacrificing  one  particle  of  interest.  The  book 
reads  like  a  romance,  and,  while  drawing  the 
student  by  an  irresistible  fascination  to  his 
task,  impresses  the  great  outlines  indelibly  upon  the  memory. 

27 


THE  NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


DRAWING. 

BARNES'S     POPULAR     DRAWING     SERIES. 

Based  upon  the  experience  of  the  most  successful  teachers  of  drawing  in  the  United 
States. 

The  Primary  Course,  consisting  of  a  manual,  ten  cards,  and  three  primary 
drawing  books.  A,  B,  and  C. 

Intermediate  Course.     Four  numbers  and  a  manual. 

Advanced  Course.    Four  numbers  and  a  manual. 

Instrumental  Course.     Four  numbers  and  a  manual. 

The  Intenneuiate,  Advanced,  and  Instrumental  Courses  are  furnished  either  in  book 
or  card  form  at  the  same  prices.  The  books  contain  the  usual  blanks,  with  the  unusual 
advantage  of  opening  Irom  the  pupil,  —  placing  the  copy  directly  in  front  and  above 
the  blank,  thus  occupying  but  little  desk-room.  The  cards  are  in  the  end  more  econom- 
ical than  the  books,  if  used  in  connection  with  the  patent  blank  folios  that  accompany 
this  series. 

The  cards  are  arranged  to  be  bound  (or  tied)  in  the  folios  and  removed  at  pleasure. 
The  pupil  at  the  end  of  each  number  has  a  complete  book,  containing  only  his  own 
work,  while  the  copies  are  preserved  and  inserted  in  another  fol.o  reaily  for  use  in  the 
next  class. 

Patent  Blank  Folios.  No.  1.  Adapted  to  Intermediate  Course.  No.  2.  Adap'ed 
to  Advanced  and  Instrumental  Courses. 

ADVANTAGES   OF  THIS   SERIES. 

The  Plan  and  Arrangement.  — The  examples  are  so  arranged  that  teachers  and 
pupils  can  see,  at  a  glance,  how  they  are  to  be  treated  and  where  they  are  to  lie  copied. 
In  this  system,  copying  and  designing  do  not  receive  all  the  attention.  The  plan  is 
broader  in  its  aims,  dealing  with  drawing  as  a  branch  of  common-school  instruction, 
and  giving  it  a  wide  educational  value. 

Correct  Methods.  —  In  this  system  the  pupil  is  led  to  rely  upon  himself,  and  not 
upon  delusive  mechanical  aids,  as  printed  guide-marks,  &c. 

One  of  the  principal  objects  of  any  good  course  in  freehand  drawing  is  to  educate  the 
eye  to  estimate  location,  form,  and  size.  A  system  which  weakens  the  motive  or  re- 
moves the  necessity  of  thinking  is  false  in  theory  and  ruinous  in  practice.  The  object 
should  be  to  educate,  not  crani  ;  to  develop  the  intelligence,  not  teach  tricks. 

Artistic  Effect.  — The  beauty  of  the  examples  is  not  destroyed  by  crowding  the 
pages  with  useless  arid  badly  printed  text.  The  Manuals  contain  all  necessary 
instruction. 

Stages  of  Development. —Many  of  the  examples  are  accompanied  by  diagrams, 
showing  the  different  stages  of  development. 

Lithographed  Examples.  — The  examples  are  printed  in  imitation  of  pencil 
drawing  (not  in  hard,  black  lines)  that  the  pupil's  work  may  resemble  them. 

One  Term's  Work.  —  Each  book  contains  what  can  be  accomplished  in  an  average 
term,  and  no  more.  Thus  a  pupil  finishes  one  book  before  beginning  another. 

Quality  —  not  Quantity.  —  Success  in  drawing  depends  upon  the  amount  of  thovght 
exercised  by  the  pupil,  and  not  upon  the  large  number  of  examples  drawn. 

Designing.  — Elementary  design  is  more  skilfully  taught  in  this  system  than  by 
any  other.  In  addition  to  the  instruction  given  in  the  books,  the  pupilwill  find  prii.ted 
on  the  insides  of  the  covers  a  variety  of  beautiful  patterns. 

Enlargement  and  Reduction.  — The  practice  of  enlarging  and  reducing  from 
copies  is  not  commenced  until  the  pupil  is  well  advanced  in  the  course  and  therefore 
better  able  to  cope  with  this  difficult  feature  in  drawing. 

Natural  Forms.  —This  is  the  only  course  that  gives  at  convenient  intervals  easy 
arid  progressive  exercises  in  the  drawing  of  natural  forms. 

Economy.  —  By  the  patent  binding  described  above,  the  copies  need  not  be  thrown 
aside  when  a  book  is  filled  out,  but  are  preserved  in  perfect  condition  for  future  use. 
The  blank  books,  only,  will  have  to  be  purchased  after  the  first  introduction,  thus  effect- 
ing a  saving  of  more  than  half  in  the  usual  cost  of  drawing-books. 

Manuals  for  Teachers.  —  The  Manuals  accompanying  this  series  contain  practical 
instructions  for  conducting  drawing  in  the  class-room,  with  definite  directions  for  draw- 
ing each  of  the  examples  in  the  books,  instructions  for  designing,  model  and  object 
drawing,  drawing  from  natural  forms,  &c. 

28 


THE  NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 

DRAWING  —  Continued. 

Chapman's  American  Drawing-Book. 

The  standard  American  text-book  and  authority  in  all  branches  of  art.  A  compilation 
of  art  principles.  A  manual  for  the  amateur,  and  basis  of  study  for  the  professional 
artist.  Adapted  for  schools  and  private  instruction. 

CONTENTS.  —  "Any  one  who  can  Learn  to  Write  can  Learn  to  Draw. "  — Primary  In- 
struction in  Drawing.  — Rudiments  of  Drawing  the  Human  '  Head.  —  Rudiments  in 
Drawing  the  Human  Figure.  —  Rudiments  of  Drawing.  —  The  Elements  of  Geometry.  - 
Perspective.  —  Of  Studying  and  Sketching  from  Nature.  — Of  Painting.  — Etching  and 
Engraving .  —  Of  Modelling.  —Of  Composition.  —  Advice  to  the  American  Art-Student. 

The  work  is  of  course  magnificently  illustrated  with  all  the  original  designs. 

Chapman's  Elementary  Drawing-Book. 

A  progressive  course  of  practical  exercise.*,  or  a  text-book  for  the  training  of  the 
eye  and  hand.  It  contains  the  elements  from  the  larger  work,  and  a  copy  should  be  in 
the  hands  of  every  pupil  ;  while  a  copy  of  the  "  American  Drawing-Book,"  named  above, 
should  be  at  hand  for  reference  by  the  class. 

Clark's  Elements  of  Drawing. 

A  complete  course  in  this  graceful  art,  from  the  first  rudiments  of  outline  to  the 
finished  sketches  of  landscape  and  scenery. 

Allen's  Map-Drawing  and  Scale. 

This  method  introduces  a  new  era  in  map-drawing,  for  the  following  reasons  :  1.  It 
is  a  system.  This  is  its  greatest  merit.  —  2.  It  is  easily  understood  and  taught.  — 
3.  The  eye  is  trained  to  exact  measurement  by  the  use  of  a  scale.  — 4.  By  no  special 
effort  of  the  memory,  distance  and  comparative  size  are  fixed  in  the  mind.  —  5.  It  dis- 
cards useless  construction  of  lines. —6.  It  can  be  taught  by  any  teacher,  even  though 
there  may  have  been  no  previous  practice  in  map-drawing.  —  7.  Any  pupil  old  enough 
to  study  geography  can  learn  by  this  system,  in  a  short  time,  to  draw  accurate  maps. 
—  8.  The  system  is  not  the  result  of  theory,  but  comes  directly  from  the  school-room. 
It  has  been  thoroughly  and  successfully  tested  there,  with  all  grades  of  pupils.  —  J>.  It 
is  economical,  as  it  requires  no  mapping  plates.  It  gives  the  pupil  the  ability  of  rapidly 
drawing  accurate  maps. 

FINE     ARTS. 

Hamerton's  Art  Essays  (Atlas  Series)  :  — 

No.  1.    The  Practical  "Work  of  Painting. 

"With  portrait  of  Rubens.     8vo.     Paper  covers. 

No.  2.    Modern  Schools  of  Art- 
Including  American,  English,  and  Continental  Painting.    8vo.     Paper  covers. 

Huntington's  Manual  of  the  Fine  Arts. 

A  careful  manual  of  instruction  in  the  history  of  art,  up  to  the  present  time. 

Boyd's  Kames'  Elements  of  Criticism. 

The  best  edition  of  the  best  work  on  art  and  literary  criticism  ever  produced  in 
English. 

Benedict's  Tour  Through  Europe. 

A  valuable  companion  for  any  one  wishing  to  visit  the  galleries  and  sights  of  the 
continent  of  Europe,  as  well  as  a  charming  book  of  travels. 

Dwight's  Mythology. 

A  knowledge  of  mythology  is  necessary  to  an  appreciation  of  ancient  art 

Walker's  World's  Fair. 

The  industrial  and  artistic  display  at  the  Centennial  Exhibition. 

29 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

BOOK-KEEPING  TEXT. 

Powers's  Practical  Book-keeping. 
Powers's  Blanks  to  Practical  Book-keeping. 

A  Treatise  on  Book-keeping,  for  Public  Schools  and  Academies.  By  Millard  R. 
Powers,  M.  A.  This  work  is  designed  to  impart  instruction  upon  the  science  of  accounts, 
as  applied  to  mercantile  business,  and  it  is  Iwlieved  that  more  knowledge,  and  thj.t,  tooj 
of  a  more  practical  nature,  can  be  gained  by  the  plan  introduced  in  this  work,  than  by 
any  other  published. 

Folsom's  Logical  Book-keeping. 
Folsom's  Blanks  to  Book-keeping. 

This  treatise  embraces  the  interesting  and  important  discoveries  of  Professor  Folsom  (of 
the  Albany  "  Bryant  &  Stratton  College  "),  the  partial  enunciation  of  which  in  lectures 
and  otherwise  has  attracted  so  much  attention  in  circles  interested  in  commercial 
education. 

After  studying  business  phenomena  for  many  years,  he  has  arrived  at  the  positive 
laws  and  principles  that  underlie  the  whole  subject  of  accounts  ;  finds  that  the  science 
is  based  in  value  as  a  generic  term  ;  that  value  divides  into  two  clisses  with  varied 
species  ;  that  all  the  exchanges  of  values  are  reducible  to  nine  equations  ;  and  that  all 
the  results  of  all  these  exchanges  are  limited  to  thirteen  in  number. 

As  accounts  have  been  universally  taught  hitherto,  without  setting  out  from  a  radical 
analysis  or  definition  of  values,  the  science  has  been  kept  in  great  obscurity,  and  been 
made  as  difficult  to  impart  as  to  acquire.  On  the  new  theory,  however,  these  obstacles 
are  chiefly  removed.  In  reading  over  the  first  part  of  it,  in  which  the  governing  laws 
and  principles  arc  discussed,  a  person  with  ordinary  intelligence  will  obtain  a  fair  con- 
ception of  the  double-entry  process  of  accounts.  But  when  he  comes  to  study  thoroughly 
these  laws  and  principles  as  there  enunciated,  and  works  out  the  examples  and  memo- 
randa which  elucidate  the  thirteen  results  of  business,  the  student  will  neither  fail  in 
readily  acquiring  the  science  as  it  is,  uor  in  becoming  able  intelligently  to  apply  it  in 
the  interpretation  of  business 

Smith  and  Martin's  Book-keeping. 
Smith  and  Martin's  Blanks. 

This  work  is  by  a  practical  teacher  and  a  practical  book-keeper.  It  is  of  a  thoroughly 
popular  class,  and  will  be  welcomed  by  every  one.  who  loves  to  see  theory  and  practice 
combined  in  an  easy,  concise,  and  methodical  form. 

The  single-entry  portion  is  well  adapted  to  supply  a  want  felt  in  nearly  all  other 
treatises,  which  seem  to  be  prepared  mainly  for  the  use  of  wholesale  merchants  ; 
leaving  retailers,  mechanics,  farmers,  &c. ,  who  transact  the  greater  portion  of  the 
business  of  the  country,  without  a  guide.  The  work  is  also  commended,  on  this 
account,  for  general  use  in  young  ladies'  seminaries,  where  a  thorough  grounding 
in  the  simpler  form  of  accounts  will  be  invaluable  to  the  future  housekeepers  of  the 
nation. 

The  treatise  on  double-entry  book-keeping  combines  all  the  advantages  of  the 
most  recent  methods  with  the  utmost  simplicity  of  application,  thus  affording  the 
pupil  all  the  advantages  of  actual  experience  in  the  counting-house,  and  giving  a 
clear  comprehension  of  the  entire  subject  through  a  judicious  course  of  mercantile 
transactions. 

PRACTICAL   BOOK-KEEPING. 

Stone's  Post-Office  Account  Book. 

By  Micah  H.  Stone.  For  record  of  Box  Rents  and  Postages.  Three  sizes  always  in 
stock.  64,  108,  and  204  pages. 

INTEREST    TABLES. 

Brooks's  Circular  Interest  Tables. 

To  calculate  simple  and  compound  interest  for  any  amount,  from  1  cent  to  §1,000,  at 
current  rates  from  1  day  to  7  years. 

31 


THE  NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 

DR.  STEELE'S  ONE-TERM  SERIES, 
IN  ALL  THE  SCIENCES. 

Steele's  i4-Weeks  Course  in  Chemistry. 
Steele's  14- Weeks  Course  in  Astronomy. 
Steele's  i4-Weeks  Course  in  Physics. 
Steele's  i4-Weeks  Course  in  Geology. 
Steele's  i4-Weeks  Course  in  Physiology. 
Steele's  i4-Weeks  Course  in  Zoology. 
Steele's  i4-Weeks  Course  in  Botany. 

Our  text-books  in  these  studies  are,  as  a  general  thing,  dull  and  uninteresting. 
They  contain  from  400  to  600  pages  of  dry  facts  and  unconnected  details.  They  abound 
in  that  which  the  student  cannot  learn,  much  less  remember.  Tlie  pupil  commences 
the  study,  is  confused  by  the  line  print  arid  coarse  print,  and  neither  knowing  exactly 
what  to  learn  nor  what  to  hasten  over,  is  crowded  through  the  single  term  generally 
assigned  to  each  branch,  and  frequently  comes  to  the  close  without  a  definite  and  exact 
idea  of  a  single  scientific  principle. 

Steele's  "  Fourteen- Weeks  Courses  "  contain  only  that  which  every  well-informed  per- 
son should  know,  while  all  that  which  concerns  only  the  professional  scientist  is  omitted. 
The  language  is  clear,  simple,  and  interesting,  and  the  illustrations  bring  the  subject 
within  the  range  of  home  life  and  daily  experience.  They  give  such  of  the  general 
principles  and  the  prominent  facts  as  a  pupil  can  make  familiar  as  household  words 
within  a  single  term.  The  type  is  large  and  open  ;  there  is  no  fine  print  to  annoy  ; 
the  cuts  are  copies  of  genuine  experiments  or  natural  phenomena,  and  are  of  fine 
execution. 

In  fine,  by  a  system  of  condensation  peculiarly  his  own,  the  author  reduces  each 
branch  to  the  limits  of  a  single  term  of  study,  while  sacrificing  nothing  that  is  essential, 
and  nothing  that  is  usually  retained  from  tire  study  of  the  larger  manuals  in  common 
use.  Thus  the  student  has  rare  opportunity  to  economize  his  time,  or  rather  to  employ 
that  which  he  has  to  the  best  advantage. 

A  notable  feature  is  the  author's  charming  "  style,"  fortified  by  an  enthusiasm  over 
his  subject  in  which  the  student  will  not  fail  to  partake.  Believing  that  Natural 
Science  is  full  of  fascination,  he  has  moulded  it  into  a  form  that  attracts  the  attention 
and  kindles  the  enthusiasm  of  the  pupil. 

The  recent  editions  contain  the  author's  "  Practical  Questions  "  on  a  plan  never 
before  attempted  in  scientific,  text-books.  These  are  questions  as  to  the  nature  and 
cause  of  common  phenomena,  and  are  not  directly  answered  in  the  text,  the  design 
being  to  test  and  promote  an  intelligent  use  of  the  student's  knowledge  of  the  foregoing 
principles. 

Steele's  Key  to  all  His  Werks. 

This  work  is  mainly  composed  of  answers  to  the  Practical  Questions,  and  solutions  of  the 
problems,  in  the  author's  celebrated  "  Fourteen-Weeks  Courses  "  in  the  several  sciences, 
with  many  hints  to  teachers,  minor  tables,  &c.  Should  be  on  every  teacher's  desk. 

Prof.  J.  Dorman  Steele  is  an  indefatigable  student,  as  well  as  author,  and  his  books 
have  reached  a  fabulous  circulation.  It  is  sate  to  say  of  his  books  that  they  have 
accomplished  more  tangible  and  better  results  in  the  class-room  than  any  other  ever 
offered  to  American  schools,  and  have  been  translated  into  more  languages  for  foreign 
schools.  They  are  even  produced  in  raised  type  for  the  blind. 

32 


THE  NATIONAL    SERIES   OF   STANDARD   SCHOOL-BOOKS. 


THE   NEW  GANOT. 

Introductory  Course  of  Natural  Philosophy. 

This  book  was  originally  edited  from  Ganot's  "  Popular  Physics,"  by  William  G. 
Peck,  LL.D.,  Professor  of  Mathematics  and  Astronomy,  Columbia  College,  and  of 
Mechanics  in  the  School  of  Mines.  It  has  recently  been  revised  by  Levi  S.  Bur- 
bank,  A.  M.,  late  Principal  of  Warren  Academy,  Woburn,  Mass.,  arid  James  I.  Hanson, 
A.M.,  Principal  of  the  High  School,  Woburn,  Mass. 

Of  elementary  works  those  of  M.  Ganot  stand  pre-eminent,  not  only  as  popular 
treatises,  but  as  thoroughly  scientific  expositions  of  the  principles  of  Physics.  His 
"  Traite  de  Physique  "  has  not  only  met  with  unprecedented  success  in  France,  but  has 
been  extensively  used  in  the  preparation  of  the  best  works  on  Physics  that  have  been 
Issued  from  the  American  press. 

In  addition  to  the  "Traite  de  Physique,"  which  is  intended  for  the  use  of  colleges 
and  higher  institutions  of  learning,  M.  Ganot  published  this  more  elementary  work, 
adapted  to  the  use  of  schools  and  academies,  in  which  he  faithfully  preserved  the 
prominent  features  and  all  the  scientific  accuracy  of  the  larger  work.  It  is  charcter- 
ized  by  a  well-balanced  distribution  of  subjects,  a  logical  development  of  scientific 
principles,  and  a  remarkable  clearness  of  definition  and  explanation.  In  addition,  it  is 
profusely  illustrated  with  beautifully  executed  engravings,  admirably  calculated  to 
convey  to  the  mind  of  the  student  a  clear  conception  of  the  principles  unfolded.  Their 
completeness  and  accuracy  are  such  as  to  enable  the  teacher  to  dispense  with  much  of 
the  apparatus  usually  employed  in  teaching  the  elements  of  Physical  Science. 

After  several  years  of  great  popularity  the  American  publishers  have  brought  this 
important  book  thoroughly  up  to  the  times.  The  death  of  the  accomplished  educator, 
Professor  Burbank,  took  place  before  he  had  completed  his  work,  and  it  was  then 
taken  in  hand  by  his  friend,  Professor  Hanson,  who  was  familiar  with  his  plans,  and 
lias  ably  and  satisfactorily  brought  the  work  to  completion. 

The  essential  characteristics  and  general  plan  of  the  book  have,  so  far  as  possible, 
been  retained,  but  at  the  same  time  many  parts  have  been  entirely  rewritten,  much 
new  matter  added,  a  large  number  of  new  cuts  introduced,  and  the  whole  treatise 
thoroughly  revised  and  brought  into  harmony  with  the  present  advanced  stage  of  sci- 
entific discovery. 

Among  the  new  features  designed  to  aid  in  teaching  the  subject-matter  are  the 
summaries  of  topics,  which,  it  is  thought,  will  be  found  very  convenient  in  short 
reviews. 

As  many  teachers  prefer  to  prepare  their  own  questions  on  the  text,  and  many  do  not 
have  time  to  spend  in  the  solution  of  problems,  it  has  been  deemed  expedient  to  insert 
both  the  review  questions  and  problems  at  the  end  of  the  volume,  to  be  used  or  not  at 
the  discretion  of  the  instructor. 


Fram  the  Churchman. 

"  No  department  of  science  has  under- 
gone so  many  improvements  and  changes 
in  the  last  quarter  of  a  century  as  that  of 
natural  philosophy.  So  many  and  so  im- 
portant have  been  the  discoveries  and 
inventions  in  every  branch  of  it  that 
everything  seems  changed  but  its  funda- 
mental principles.  Ganot  has  chapter 
upon  chapter  upon  subjects  that  were  not 
so  much  as  known  by  name  to  Olmsted  ; 
and  here  we  have  Ganot,  first  edited  by 
Professor  Peck,  and  afterward  revised  by 
the  late  Mr.  Burbank  and  Mr.  Hanson.  No 
elementary  works  upon  philosophy  have 
been  superior  to  those  of  Ganot,  either  as 
popular  treatises  or  as  scientific  exposi- 
tions of  the  principles  of  physics,  and 
his  '  Traite  de  Physique '  has  not  only  had 
a  great  success  in  France,  but  has  been 
freely  used  in  this  country  in  the  prepa- 
ration of  American  books  upon  the  sub- 


jects of- which  it  treats.  That  work  was 
intended  for  higher  institutions  of  learn- 
ing, and  Mr.  Ganot  prepared  a  more 
elementary  work  for  schools  and  acade- 
mies. It  is  as  scientifically  accurate  as 
the  larger  work,  and  is  characterized  by 
a  logical  development  of  scientific  princi- 
ples, by  clearness  of  definition  and  expla- 
nation, by  a  proper  distribution  of  sub- 
jects, and  by  its  admirable  engravings. 
We  here  have  Ganot's  work  enhanced  in 
value  by  the  labors  of  Professor  Peck  and  of 
Messrs.  Burbank  and  Hanson,  and  brought 
up  to  our  own  times.  The  essential  char- 
acteristics of  Ganot's  work  have  been  re- 
tained, but  much  of  the  book  lias  been 
rewritten,  and  many  new  cuts  have  been 
introduced,  made  necessary  by  the  prog- 
ress of  scientific  discovery.  The  short 
reviews,  the  questions  on  the  text,  and 
the  problems  given  for  solution  are  desir- 
able additions  to  a  work  of  this  kind,  and 
will  give  the  book  increased  popularity." 


34 


THE  NATIONAL   SERIES  OF  STANDARD  SCHOOL-BOOKS. 


FAMILIAR    SCIENCE. 

Norton  &  Porter's  First  Book  of  Science. 

Setsf>rth  the  principles  of  Natural  Philosophy,  Astronomy,  Chemistry,  Physic  "o  y. 
anJ  Geology,  on  the  catechetical  plan  for  primary  classes  an  I  beginners. 

Chambers's  Treasury  of  Knowledge. 

Progressive  lessons  upon — first,  common  things  which  lie  most  immediately  around 
us,  and  first  attract  the  attention  of  the  young  mind  ;  second,  common  objects  from  ti.e 
mineral,  animal,  and  vegetable  kingdoms,  manufactured  articles,  and  miscellaneous 
substances  ;  third,  a  systematic  view  of  nature  under  the  various  sciences.  May  be 
used  as  a  reader  or  text-book. 

Monteith's  Easy  Lessons  in  Popular  Science. 

This  book  combines  within  its  covers  more  attractive  features  for  the  study  of  science 
by  children  than  any  other  book  published.  It  is  a  reading  book,  spelling  book,  com- 
position book,  drawing  book,  geography,  history,  book  on  botany,  zoology,  agricul- 
ture, manufactures,  commerce,  and  natural  philosophy.  All  these  subjects  are  presented 
in  a  simple  and  effective  style,  such  as  would  be  adopted  by  a  good  teacher  on  an 
excursion  with  a  class.  The  class  are  supposed  to  be  taking  excursions,  with  the  help 
of  a  large  pictorial  chart  of  geography,  which  can  be  suspended  before  them  in  the 
school-room.  A  key  of  the  chart  is  inserted  in  every  copy  of  the  book.  With  this 
book  the  science  of  common  or  familiar  things  can  be  taught  to  beginners. 


NATURAL    PHILOSOPHY. 

Norton's  First  Book  in  Natural  Philosophy. 
Peck's  Elements  of  Mechanics. 

A  suitable  introduction  to  Bartlett's  higher  treatises  on  Mechanical  Philosophy,  and 
adequate  in  itself  tor  a  complete  academical  course. 

Bartlett's  Analytical  Mechanics. 
Bartlett's  Acoustics  and  Optics. 

A  complete  system  of  Collegiate  Philosophy,  by  Prof.  W.  H.  C.  Bartlett,  of  West 
Point  Military  Academy. 

Steele's  Physics. 

Peck's  Ganot. 

GEOLOGY. 

Page's  Elements  of  Geology. 

A  volume  of  Chambers's  Educational  Course.  Practical,  simple,  an/1  "tninently 
calculated  to  make  the  study  interesting. 

Steele's  Geology. 

CHEMISTRY. 

Porter's  First  Book  of  Chemistry. 
Porter's  Principles  of  Chemistry. 

The  above  are  widely  known  as  the  productions  of  one  of  the  most  eminent  scientific 
men  of  America.  The  extreme  simplicity  in  the  method  of  presenting  the  science,  while 
exhaustively  treated,  lias  excited  universal  commendation. 

Gregory's  Chemistry  (Organic  and  Inorganic).     2  vols, 

The  science  exhaustively  treated.     For  colleges  and  medical  students. 

Steele's  Chemistry. 

36 


THE  NATIONAL   SERIES   OF  STANDARD   SCHOOL-BOOKS. 

NATURAL    SCIENCE  —  Continued. 

BOTANY. 

Wood's  Object-Lessons  in  Botany. 
Wood's  American  Botanist  and  Florist. 
Wood's  New  Class-Book  of  Botany. 

The  standard  text-hooks  of  the  United  States  in  this  department.  In  style  they  are 
simple,  popular,  and  lively  ;  in  arrangement,  easy  and  natural  ;  in  description,  graphic 
;:nd  scientific.  The  Tables  for  Analysis  are  reduced  to  a  perfect  system.  They  include 
the  flora  ol  the  whole  United  States  east  of  the  Rocky  Mountains,  and  are  well  adapted 
to  the  regions  west. 

Wood's  Descriptive  Botany. 

A  complete  flora  of  all  plants  growing  east  of  the  Mississippi  River. 

Wood's  Illustrated  Plant  Record. 

A  simple  form  of  blanks  for  recording  observations  in  the  field. 

Wood's  Botanical  Apparatus. 

A  portable  trunk,  containing  drying  pre.',s,  knife,  trowel,  microscope,  and  tweezers, 
and  a  copy  of  Wood's  "  Plant  Record,"  —  the  collector's  complete  outlit. 

Willis's  Flora  of  New  Jersey. 

The  most  useful  book  of  reference  e<cr  published  for  collectors  in  all  parts  of  the 
country.  It  contains  also  a  Botanical  Directory,  with  addresses  of  living  American 
botanists. 

Young's  Familiar  Lessons  in  Botany. 

Combining  simplicity  of  diction  with  some  degree  of  technical  and  scientific  knowl- 
edge, lor  intermediate  classes.  Specially  adapted  for  the  Southwest. 

Wood  &  Steele's  Botany. 

See  page  33. 


AGRICULTURE. 

Pendleton's  Scientific  Agriculture. 

A  text-book  for  colleges  and  schools  ;  treats  of  the  following  topics :  Anatomy  and 
Physiology  of  Plants  ;  Agricultural  Meteorology  ;  Soils  as  related  to  Physics  ;  Chemistry 
of  the  Atmosphere  ;  of  Plants  ;  of  Soils  ;  Fertilizers  and  Natural  Manures;  Animal  Nu- 
trition, &c.  I3y  E.  M.  Pendleton,  M.  D.,  Professor  of  Agriculture  in  the  University  of 
Georgia. 


From  PRESIDENT  A.  D.  WHITE,  Cornell 

University. 

"  Dear  Sir :  I  have  examined  your 
'  Text-book  of  Agricultural  Science,'  and  it 
seems  to  me  excellent  in  view  of  the  pur- 
pose it  is  intended  to  serve.  Many  of 
your  chapters  interested  me  especially, 
and  all  parts  of  the  work  seem  to  combine 
scientific  instruction  with  practical  infor- 
mation in  proportions  dictated  by  sound 
common  sense." 


From    PRESIDENT    ROBINSON,    of  Brown 

University. 

"  It  is  scientific  in  method  as  well  as  in 
matter,  comprehensive  in  plan,  natural 
and  logical  in  order,  compart  and  lucid  in 
its  statements,  and  must.be  useful  both  as 
a  text-book  in  agricultural  colleges,  and 
as  a  hand-book  for  intelligent  planters  and 
farmers." 


37 


THE  NATIONAL   SER/ES   OF  STANDARD   SCHOOL-BOOKS. 

NATURAL  SCIENCE— Continued. 

PHYSIOLOGY. 

Jarvis's  Elements  of  Physiology. 
Jarvis's  Physiology  and  Laws  of  Health. 

The  only  books  extant  \vhicn  approach  this  subject  with  a  proper  view  of  the  true 
object  of  teaching  Physiology  iu  schools,  viz.,  that  scholars  may  know  how  to  take  care 
of  their  own  health.  In  bold  contrast  with  the  abstract  Anatomies,  which  children 
learn  as  they  would  Greek  or  Latin  (and  forget  as  soon),  to  discipline  the  mind,  are  these 
text-books,  using  the  science  as  a  secondary  consideration,  and  only  so  far  as  is  neces- 
sary for  the  comprehension  of  the  laws  o,  tiealtk. 

Steele's  Physiology. 

See  page  33.  


ASTRONOMY. 

Willard's  School  Astronomy. 

J3y  means  of  clear  and  attractive  illustrations,  addressing  the  eye  in  many  cases  by 
analogies,  careful  definitions  of  all  necessary  technical  terms,  a  careful  avoidance  of  ver- 
biage, and  unimportant  matter,  particular  attention  to  analysis,  and  a  general  .adoption 
of  the  simplest  methods,  Mrs.  Willard  has  made  the  best  and  most  attractive  elemen- 
tary Astronomy  extant. 

Mclntyre's  Astronomy  and  the  Globes. 

A  complete  treatise  for  intermediate  classes.     Highly  approved. 

Bartlett's  Spherical  Astronomy. 

The  West  Point  Course,  for  advanced  classes,  with  applications  to  the  current  wants 
of  Navigation,  Geography,  and  Chronology. 

Steele's  Astronomy. 

See  page  33.  . 

NATURAL    HISTORY. 

Carll's  Child's  Book  of  Natural  History. 

Illustrating  the  animal,  vegetable,  and  mineial  kingdoms,  witli  application  to  the 
arts.  For  beginners.  Beautifully  ai,d  copiously  illustrated. 

Anatomical  Technology.     Wilder  &  Gage. 

As  applied  to  the  domestic  cat.     For  the  use  of  students  of  medicine. 


ZOOLOGY. 

Chambers's  Elements  of  Zoology. 

A  complete  and  comprehensive  system  of  Zoology,  adapted  for  academic  instruction, 
presenting  a  systematic  view  of  the  animal  kingdom  as  a  portion  of  external  nature. 

Steele's  Zoology. 

See  page  33. 

38 


THE  NATIONAL   SERIES  OF  STANDARD   SCHOOL-BOOKS. 


LITERATURE. 

Oilman's  First  Steps  in  English  Literature. 

The  character  and  plan  of  this  exquisite  little  text-book  may  be  best  understood  irom 
an  analysis  of  its  contents  :  Introduction.  Historical  Period  of  Immature  English, 
with  Chart ;  Definition  of  Terms  ;  Languages  of  Europe,  with  Chart ;  Period  of  Mature 
English,  with  Chart  ;  a  Chart  of  Bible  Translations,  a  Bibliography  or  Guide  to  General 
Reading,  and  other  aids  to  the  student. 

Cleveland's  Compendiums.     3  vols.     12mo. 

ENGLISH  LITERATURE.  AMERICAN  LITERATURE. 

ENGLISH  LITERATURE  OF  THE  XIXTii  CENTURY. 

In  these  volumes  are  gathered  the  cream  of  the  literature,  of  the  English-speaking 
people  for  the  school-room  and  the  general  reader.  Their  reputation  is  national.  More 
than  125,000  copies  have  been  sold. 

Boyd's  English  Classics.     6  vols.    Cloth.    12mo. 
MILTON'S  PARADISE  LOST.  THOMSON'S  SEASONS 

YOUNG'S  NIGHT  THOUGHTS.  FOLLOK'S  COURSE  OF  TIME. 

COWPER'S  TASK,  TABLE  TALK,  &c.    LORD  BACON'S  ESSXYS. 

This  series  of  annotated  editions  of  great  English  writers  in  prose  and  poetry  is 
designed  for  critical  reading  and  parsing  in  schools.  Prof.  J.  R.  Boyd  proves  himself 
an  editor  of  high  capacity,  and  the  works  themselves  need  no  encomium.  As  auxiliary 
to  the  study  of  belles-lettres,  &e.,  these  works  have  no  equal. 

Pope's  Essay  on  Man.     Ifimo.  Paper. 
Pope's  Homer's  Iliad.     32mo.  Roan. 

The  metrical  translation  of  the  great  poet  of  antiquity,  and  the  matchless  "Essay  on 
the  Nature  and  State  of  Man,"  by  Alexander  Pope,  afford  superior  exercise  in  literature 
and  parsing. 


POLITICAL    ECONOMY. 

Champlin's  Lessons  on  Political  Economy. 

An  improvement  on  previous  treatises,  being  shorter,  yet  containing  evorythim> 
essential,  with  a  view  of  recent  questions  in  finance,  &c.,  which  is  not  elsewhere 
found. 

39 


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a. 


£-- 


